![Preface vii
and Laurent, and which will be used in the subsequent chapters. In all but one of these
estimates, ˇ1; : : : ; ˇn are integers, a special case sufficient for most of the applications.
The lower bounds are then expressed in terms of the maximum B of their absolute values
and take the form
log jƒnj c.n; D/ .log 2A1/ : : : .log 2An/ .log 2B/;
where c.n; D/ is an explicit real number depending only on n and the degree D of the
algebraic number field generated by ˛1; : : : ; ˛n and Aj is the maximum of the absolute
values of the coefficients of the minimal defining polynomial of ˛j over the rational
integers, for j D 1; : : : ; n. The crucial achievements of Baker are the logarithmic
dependence on B and the fact that an admissible value for c.n; D/ can be explicitly
computed.
We consider in Chapter 3 Diophantine problems for which the reduction to linear
forms in complex logarithms is almost straightforward. These problems include explicit
lower bounds for the distance between powers of 2 and powers of 3, effective irrationality
measures for n-th roots of rational numbers, lower bounds for the greatest prime factor
of n.n C 1/, where n is a positive integer, perfect powers in linear recurrence sequences
of integers, etc.
Chapter 4 is devoted to applications to classical families of Diophantine equations. In
the works of Thue and Siegel, it was established that unit equations, Thue equations, and
super- and hyperelliptic equations have only finitely many integer solutions, but the proofs
were ineffective, in the sense that they did not yield upper bounds for the absolute values
of the solutions and, consequently, were of very little help for the complete resolution
of the equations. The theory of linear forms in logarithms induced dramatic changes in
the field of Diophantine equations and we explain how it can be applied to establish, in
an effective way, that unit equations, Thue equations, super- and hyperelliptic equations,
the Catalan equation, etc., have only finitely many integer solutions. This chapter also
contains a complete proof, following Bilu and Bugeaud [72], of an effective improvement
of Liouville’s inequality (which states that an algebraic number of degree d cannot be
approximated by rational numbers at an order greater than d) derived ultimately from an
estimate for linear forms in two complex logarithms proved in Chapter 11.
When the algebraic numbers ˛1; : : : ; ˛n occurring in the linear form ƒn are all rational
numbers very close to 1, the lower bounds for jƒnj can be considerably improved. Several
applications of this refinement are listed in Chapter 5. They include effective irrationality
measures for n-th roots of rational numbers close to 1 and striking results on the Thue
equation axn
byn
D c.
Chapter 6 presents various applications of the theory of linear forms in p-adic log-
arithms, in particular towards Waring’s problem and, again, to perfect powers in linear
recurrence sequences of integers. It also includes extensions of results established in
Chapter 4: unit equations, Thue equations, super- and hyperelliptic equations have only
finitely many solutions in the rational numbers, whose denominators are divisible by
prime numbers from a given, finite set, and, moreover, the size of these solutions can be
effectively bounded.
Primitive divisors of terms of binary recurrence sequences are discussed in Chapter 7.
We partially prove a deep result of Bilu, Hanrot, and Voutier [77] on the primitive](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-9-320.jpg)
![viii Preface
divisors of Lucas and Lehmer numbers and discuss some of its applications to Diophantine
equations. Then, following Stewart [400], we confirm a conjecture of Erdős and show
that, for every integer n 3, the greatest prime factor of 2n
1 exceeds some positive
real number times n
p
log n= log log n.
In Chapter 8, we follow Stewart and Yu [405] to establish partial results towards the
abc-conjecture, which claims that, for every positive real number , there exists a positive
real number ./, depending only on , such that, for all coprime, positive integers a; b;
and c with a C b D c, we have
c ./
Y
pjabc
p
1C
;
the product being taken over the distinct prime factors of abc. Specifically, we show
how to combine complex and p-adic estimates to prove the existence of an effectively
computable positive real number such that, for all positive coprime integers a; b; and c
with a C b D c, we have
log c
Y
pjabc
p
1=3
log
Y
pjabc
p
3
:
There are only a few known applications of the theory of simultaneous linear forms in
logarithms, developed by Loxton in 1986. Two of them are presented in Chapter 9. A first
gives us an upper bound for the number of perfect powers in the interval ŒN; N C
p
N,
for every sufficiently large integer N. A second shows that, under a suitable assumption,
a system of two Pellian equations has at most one solution.
Given a finite set of multiplicatively dependent algebraic numbers, we establish in
Chapter 10 that these numbers satisfy a multiplicative dependence relation with small
exponents. A key ingredient for the proof is a lower bound for the Weil height of a
non-zero algebraic number which is not a root of unity.
Full proofs of estimates for linear forms in two complex logarithms, which, in partic-
ular, imply lower estimates for the difference between integral powers of real algebraic
numbers, are given in Chapter 11. Analogous estimates for linear forms in two p-adic
logarithms, that is, upper estimates for the p-adic valuation of the difference between in-
tegral powers of algebraic numbers are given in Chapter 12. An estimate for linear forms
in an arbitrary number of complex logarithms is derived in Chapter 4 from the estimate
for linear forms in two complex logarithms established in Chapter 11. While the former
estimate is not as strong and general as the estimates stated in Chapter 2, it is sufficiently
precise for many applications.
We collect open problems in Chapter 13. The thirteen chapters are complemented by
six appendices, which, mostly without proofs, gather classical results on approximation
by rational numbers, the theory of heights, algebraic number theory, and p-adic analysis.
We have tried, admittedly without too much success, to curb our taste for extensive
bibliographies. No effort has been made towards exhaustivity, including in the list of
bibliographic references, and the topics covered in this textbook reflect somehow the
personal taste of the author.](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-10-320.jpg)
![Preface ix
Inevitably, there is some overlap between this monograph and the monograph [376]
of Shorey and Tijdeman, which, although over thirty years old, remains an invaluable
reference for anyone interested in Diophantine equations. In particular, the content of
Chapter 4 (except Section 4.1) is treated in [376] in much greater generality. There is also
some overlap with Sprindžuk’s book [386] and the monograph of Evertse and Győry [182].
Regarding the theory of linear forms in logarithms, Chapters 2 and 11 can be seen as an
introduction to the book of Waldschmidt [432]. As far as we are aware, the content of
Chapters 5, 7, 8, 9, and 12 and several other parts of the present monograph have never
appeared in books.
To keep this book reasonably short and accessible to graduate and post-graduate
students, the results are not proved in their greatest generality and proofs of the best
known lower bounds for linear forms in an arbitrary number of complex (resp., p-adic)
logarithms are not given.
Many colleagues sent me comments, remarks, and suggestions. I am grateful to all of
them. Special thanks are due to Samuel Le Fourn, who very carefully read the manuscript
and sent me many insightful suggestions.
This book was written while I was director of the ‘Institut de Recherche Mathématique
Avancée’.](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-11-320.jpg)
![Chapter 1
Brief introduction to linear forms in logarithms
We start this textbook with a concise introduction to the theory of linear forms in
logarithms. For much more comprehensive historical surveys and many bibliographic
references, the reader is directed to the introductory text [441] and the monographs
[28, 38, 141, 243, 419, 432]. Throughout, unless otherwise specified, log z denotes the
principal determination of the logarithm of the non-zero complex number z, that is, writ-
ing z D rei
with r positive and in . ; , we have log z D log r C i. In particular,
log. 1/ is equal to i.
1.1. Linear forms in complex logarithms
A complex number is called algebraic if it is a root of a non-zero polynomial with integer
coefficients. A complex number which is not algebraic is called transcendental. Rational
numbers and their integer roots are obvious examples of algebraic numbers. The first
examples of transcendental numbers were given, and even in a totally explicit form, by
Liouville [264] in 1844, in a note where he proved that a real algebraic irrational number
cannot be too close, in a suitable sense, to rational numbers; see Theorem A.5 for a precise
formulation.
It is one thing to construct transcendental numbers; it is another, much more difficult
one, to prove the transcendence of some explicitly given complex numbers. The first
result in this direction is the proof of the transcendence of e, obtained by Hermite in 1873.
Shortly thereafter, in 1882, Lindemann established that is transcendental. The now
famous Hermite–Lindemann theorem reads as follows.
Theorem 1.1. For any non-zero complex number ˇ, at least one of the two numbers ˇ
and eˇ
is transcendental.
Theorem 1.1 includes the transcendence of e (take ˇ D 1) and that of (take ˇ D i).
It also shows that log 2 is transcendental and, more generally, that any determination of
the logarithm of an algebraic number different from 0 and 1 is transcendental.
In 1885 Weierstrass extended Theorem 1.1 as follows.
Theorem 1.2. Let n 2 be an integer. Let ˛1; : : : ; ˛n be distinct algebraic numbers.
Then, e˛1
; : : : ; e˛n
are linearly independent over the field of algebraic numbers.
Sincemonomialsine˛1
; : : : ;e˛n
areexponentialsofintegralcombinationsof ˛1; : : : ; ˛n,
it is easy to show that Theorem 1.2 is equivalent to the following statement: If n 2](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-19-320.jpg)
![2 1. Brief introduction to linear forms in logarithms
and ˛1; : : : ; ˛n are algebraic numbers that are linearly independent over the field of ratio-
nal numbers, then e˛1
; : : : ; e˛n
are algebraically independent over the field of algebraic
numbers (this means that no non-zero polynomials with algebraic coefficients vanish at
.e˛1
; : : : ; e˛n
/).
In 1900, David Hilbert proposed a list of twenty-three open problems and presented ten
of them in Paris at the second conference of the International Congress of Mathematicians.
His seventh problem is the following (observe that e
D . 1/ i
):
The expression ˛ˇ
for an algebraic base ˛ different from 0 and 1 and an irrational
algebraic exponent ˇ, e.g. the number 2
p
2
or e
, always represents a transcendental or
at least an irrational number.
This problem was solved in 1934 independently and simultaneously by Gelfond [197]
and Schneider [359], by different methods.
Theorem 1.3. For any non-zero algebraic numbers ˛1; ˛2; ˇ1; ˇ2 with log ˛1 and log ˛2
linearly independent over the rationals, we have
ˇ1 log ˛1 C ˇ2 log ˛2 6D 0:
Theorem 1.3 was generalized to linear combinations of n logarithms of algebraic
numbers by Baker [16,17] in 1966 and 1967.
Theorem 1.4. Let n 2 be an integer. Let ˛1; : : : ; ˛n be non-zero algebraic numbers
and log any fixed determination of the logarithm function. If log ˛1; : : : ; log ˛n are
linearly independent over the rationals, then they are linearly independent over the field
of algebraic numbers.
Shortly thereafter, Baker [18] extended his previous result as follows.
Theorem 1.5. Let n be a positive integer. Let ˛1; : : : ; ˛n be non-zero algebraic numbers
and log ˛1; : : : ; log ˛n any determinations of their logarithms. If log ˛1; : : : ; log ˛n are
linearly independent over the rationals, then 1; log ˛1; : : : ; log ˛n are linearly independent
over the field of algebraic numbers.
It readily follows from Theorem 1.5 that the complex number eˇ0
˛ˇ1
1 ˛ˇn
n is tran-
scendental for all non-zero algebraic numbers ˛1; : : : ; ˛n, ˇ0; : : : ; ˇn. Furthermore,
Theorem 1.5 includes Theorem 1.1.
Theorems 1.3 to 1.5 show that any expression of the form
ˇ0 C ˇ1 log ˛1 C C ˇn log ˛n; (1.1)
where ˛1; : : : ; ˛n, ˇ1; : : : ; ˇn are non-zero algebraic numbers and ˇ0 is algebraic, vanishes
only in trivial cases. A natural question is then to bound from below its absolute value
(when non-zero).
For the sake of simplification, we assume in the discussion below that the algebraic
numbers involved are all rational numbers. Let n 2 be an integer. For j D 1; : : : ; n,
let xj
yj
be a non-zero rational number, bj a non-zero integer, and set
B WD maxf3; jb1j; : : : ; jbnjg and Aj WD maxf3; jxj j; jyj jg: (1.2)](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-20-320.jpg)
![1.1. Linear forms in complex logarithms 3
We consider the rational number
ƒ WD
x1
y1
b1
xn
yn
bn
1: (1.3)
Since we wish to bound jƒj from below, we may assume that jƒj 1
2
. Then, the linear
form in logarithms of rational numbers , defined by
WD log.1 C ƒ/ D b1 log
x1
y1
C C bn log
xn
yn
;
satisfies
jƒj
2
jj 2jƒj:
A trivial estimate of the denominator of (1.3) gives that ƒ D 0 or
log jƒj
n
X
jD1
jbj j log maxfjxj j; jyj jg B
n
X
jD1
log Aj : (1.4)
The dependence on the Aj ’s in (1.4) is very satisfactory, unlike the dependence on B. For
applications to Diophantine problems, we require a better estimate in terms of B than the
one given in (1.4), even if it comes with a weaker one in terms of the Aj ’s. For example,
replacing B by o.B/ is sufficient in many cases (see, for example, Theorems 3.10, 3.13,
and 4.9), but not in all cases (see, for example, Theorems 3.3, 3.4, and 5.1). The next
lemma shows that B cannot be replaced by o.log B/.
Lemma 1.6. Let n; a1; : : : ; an be integers, all of which are greater than or equal to 2.
Set A D maxfa1; : : : ; ang. Then, for every integer B greater than 2n log A, there exist
rational integers b1; : : : ; bn with
0 maxfjb1j; : : : ; jbnjg B
and
jab1
1 abn
n 1j
2n log A
Bn 1
:
Lemma 1.6 is a direct consequence of the Dirichlet Schubfachprinzip applied to the
points b1 log a1 C C bn log an with 0 b1; : : : ; bn B, which all lie in the interval
Œ0; nB log A.
The first effective improvement of (1.4) was obtained by Gelfond [198] in 1935 in the
case n D 2. He proved that, for multiplicatively independent positive rational numbers
x1
y1
; x2
y2
, for an arbitrary positive real number , and for all integers b1; b2, not both 0, we
have ˇ
ˇ
ˇ
x1
y1
b1
x2
y2
b2
1
ˇ
ˇ
ˇ x1
y1
;
x2
y2
; exp .log B/5C
;
where B D maxf3; jb1j; jb2jg. Throughout this book, the notation a c1;:::;cn
b
(resp., a ineff
c1;:::;cn
b) means that the quantity a is greater than b times an effectively
computable positive real number (resp., a positive real number), which depends at most](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-21-320.jpg)
![4 1. Brief introduction to linear forms in logarithms
on c1; : : : ; cn. Subsequently, Gelfond improved his own result and totally explicit estimates
were provided by Schinzel [354] in 1967.
Gelfond [200] also gave an estimate valid for a linear form in an arbitrary number of
logarithms.
Theorem 1.7. Let n 2 be an integer and a1; : : : ; an positive rational numbers which are
multiplicatively independent. Let ı be a positive real number. Let b1; : : : ; bn be rational
integers, not all zero, and set B D maxf3; jb1j; : : : ; jbnjg. Then, we have
jab1
1 abn
n 1j ineff
n;a1;:::;an; ı exp. ıB/:
The proof of Theorem 1.7, which rests on a theorem of Siegel, does not enable us to
compute effectively the implicit numerical constant. In his book Gelfond [200] pointed
out the importance of getting an effective version of Theorem 1.7.
For n 3, the first non-trivial effective lower bound for the quantity ƒ defined in (1.3)
was given by Baker [16] in 1966. With B as in (1.2), he obtained that either ƒ D 0 or, for
every positive real number , we have
ˇ
ˇ
ˇ
x1
y1
b1
xn
yn
bn
1
ˇ
ˇ
ˇ n;
x1
y1
;:::; xn
yn
; exp .log B/nC1C
:
His result is much more general and applies to expressions of the form (1.1).
Shortly thereafter, Feldman [185,186] established the following refinement of Baker’s
lower bound.
Theorem 1.8. Let n 2 be an integer and a1; : : : ; an positive rational numbers which
are multiplicatively independent. Let b1; : : : ; bn be rational integers, not all zero, and
set B D maxf3; jb1j; : : : ; jbnjg. Then, there exists a positive, effectively computable real
number C, depending only on n; a1; : : : ; an, such that
jab1
1 abn
n 1j exp. C log B/ D B C
: (1.5)
Lemma 1.6 shows that the dependence on B in the estimate (1.5) is essentially best
possible, but, for applications, it is much desirable to determine precisely how C depends
on the rational numbers a1; : : : ; an.
A first explicit result in this direction was given by Baker [19] in 1968. Throughout
this chapter, the height of an algebraic number is the naïve height, that is, the maximum of
the absolute values of the coefficients of its minimal defining polynomial over the integers.
We reproduce below a consequence of the main theorem of [19].
Theorem 1.9. Assume that n 2 and that ˛1; : : : ; ˛n are non-zero algebraic numbers,
whose degrees do not exceed D and whose heights do not exceed A, where D 4 and
A 4. Let ı be a real number with 0 ı 1. Let b1; : : : ; bn be rational integers, not
all zero, and set B D maxf3; jb1j; : : : ; jbnjg. If
j˛b1
1 ˛bn
n 1j
e ınB
2
;
then
B .4.nC1/2
ı 1
D2.nC1/
log A/.2nC3/2
:](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-22-320.jpg)
![1.1. Linear forms in complex logarithms 5
In many applications, especially to classical families of Diophantine equations, we are
led to bound from below expressions of the form j˛b1
1 ˛bn
n ˛nC1 1j, where ˛nC1 has a
large height; see Chapter 4. In this respect, Theorem 1.9 is not plainly satisfactory, since
˛1; : : : ; ˛n play the same rôle, irrespective of their height and their exponent. Its following
refinement, established by Baker [26], appears to be very useful.
Theorem 1.10. Assume that n 1 and let ˛1; : : : ; ˛nC1 be non-zero algebraic numbers
of degree at most D. Let the heights of ˛1; : : : ; ˛n and ˛nC1 be at most A and AnC1
respectively, where A 2 and AnC1 2. Let ı be a positive real number. Let b1; : : : ; bn
be integers, not all zero, and set B D maxf3; jb1j; : : : ; jbnjg. If
0 j˛b1
1 ˛bn
n ˛nC1 1j e ıB
;
then
B n;D;A;ı log AnC1:
In the present textbook, we give a complete proof of Theorem 1.10. We derive it from
a lower bound for linear forms in two logarithms; see Section 4.1.
In 1975, Baker [27] established a subsequent refinement of Theorem 1.8.
Theorem 1.11. Assume that n 2 and that ˛1; : : : ; ˛n are non-zero algebraic numbers,
whose degrees do not exceed D. Assume that, for j D 1; : : : ; n, the height of ˛j does
not exceed Aj , where Aj 2. Let b1; : : : ; bn be rational integers, not all zero, and set
B D maxf3; jb1j; : : : ; jbnjg. If ˛b1
1 ˛bn
n is not equal to 1, then
j˛b1
1 ˛bn
n 1j B C… log …
; (1.6)
where
… D log A1 log An
and C is an effectively computable real number depending only on n and D.
As was pointed out by Baker below the statement of his theorem, it would be of much
interest to eliminate log … in (1.6). Also, Theorem 1.11 does not include Theorems 1.9
and 1.10, nor the main result of Baker’s paper [25].
The deletion of the log … factor (which is mainly interesting from a theoretical point of
view, but also has several applications, see, for example, Theorem 3.6) has been achieved
independently by Wüstholz [440] and by Philippon and Waldschmidt [329]. Thus, at the
end of the ’80s, it was established that, with ƒ, A1; : : : ; An, and B as in (1.2) and (1.3),
there exists an effectively computable real number c.n/, depending only on the number n
of rational numbers involved in (1.3), such that the lower estimate
log
ˇ
ˇ
ˇ
x1
y1
b1
xn
yn
bn
1
ˇ
ˇ
ˇ c.n/ log A1 log An log B (1.7)
holds, when ƒ is non-zero. Since then, several authors have managed to considerably
reduce the value of the real number c.n/. However, it remains an open problem to replace
the product log A1 log An by the sum log A1 C C log An; see Chapter 13.](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-23-320.jpg)
![6 1. Brief introduction to linear forms in logarithms
1.2. Linear forms in p-adic logarithms
Let p be a prime number. In parallel with the development of the theory of linear forms
in complex logarithms, progress has been regularly made towards its p-adic analogue.
For a non-zero rational number x, let vp.x/ denote the exponent of p in the decompo-
sition of x as a product of prime powers. With ƒ; A1; : : : ; An, and B as in (1.2) and (1.3),
a trivial estimate shows that
vp
x1
y1
b1
xn
yn
bn
1
log.2Ab1
1 Abn
n /
log p
1 C
B
log p
n
X
j D1
log Aj : (1.8)
The theory of p-adic linear forms in logarithms gives a much better result in terms of the
dependence on B, namely, that there exists an effectively computable real number c0
.n/,
depending only on the number n of rational numbers involved, such that the upper estimate
vp
x1
y1
b1
xn
yn
bn
1
c0
.n/ p log A1 log An log B (1.9)
holds, when ƒ is non-zero. In terms of B, this is much better than (1.8) and analogous
to (1.7). One of the major open problems in the theory is to remove (or at least to improve)
the dependence on p in (1.9), which remains very unsatisfactory and is ultimately a
consequence of the fact that the radius of convergence of the p-adic exponential function
is finite.
A good reference for an historical introduction is [448]. Mahler [278] established
in 1932 the p-adic analogue of the Hermite–Lindemann theorem and three years later [280]
the p-adic analogue of the Gelfond–Schneider theorem. Gelfond [199] proved a quan-
titative estimate for linear forms in two p-adic logarithms, which was later refined by
Schinzel [354].
Estimates for linear forms in an arbitrary number of p-adic logarithms were obtained by
Brumer [99], Sprindžuk [384,385], Coates [149], Kaufman [238], Baker and Coates [34],
van der Poorten [335], Dong [168,169], and Yu [443,445–447], among others.
1.3. Linear forms in elliptic logarithms
A few years after the birth of the theory of linear forms in logarithms, it was realized
that the techniques used in the proofs can be applied to any commutative algebraic group.
Initial steps towards the derivation of elliptic analogues were made by Baker [23], who
considerably extended earlier results of Schneider [360]. Elliptic curves with complex mul-
tiplication were studied by Masser [284] in 1975 and, a year later, Coates and Lang [151]
established the first lower bounds in the case of Abelian varieties with complex multi-
plication. Subsequently, Philippon and Waldschmidt [330] and Hirata-Kohno [229,230]
obtained rather general statements.
In the case of elliptic logarithms, the first totally explicit estimates were given by
David [162]. These have applications to the complete determination of the integral points
on an elliptic curve, as is very well explained by Stroeker and Tzanakis [408], Gebel,
Pethő, and Zimmer [196], and Tzanakis [420,421].](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-24-320.jpg)
![1.3. Linear forms in elliptic logarithms 7
Let n be a positive integer and E1; : : : ; En elliptic curves over an algebraic number
field K. For j D 1; : : : ; n, consider a Weierstrass model of Ej and }j the associated
Weierstrass function. Let uj be a complex number such that }j .uj / is in K [ f1g.
Such a complex number uj is an elliptic logarithm of an algebraic point of Ej . Also,
let ˇ0; ˇ1; : : : ; ˇn be elements of K. Let B 3 denote an upper bound for the heights
of ˇ0; ˇ1; : : : ; ˇn.
Set
ƒe WD ˇ0 C ˇ1u1 C C ˇnun:
David and Hirata-Kohno [163] established that there exists an effectively computable
positive real number c, which depends only on K; n; u1; : : : ; un and the curves E1; : : : ; En,
such that we have jƒej B c
if ƒe is non-zero.
For the state-of-the-art and generalisations to Abelian varieties, the reader is directed
to [163] and to Gaudron’s papers [192–194].
Lower bounds for linear forms in p-adic elliptic logarithms were given by Rémond
and Urfels [345] and Hirata-Kohno and Takada [232]; see also Fuchs and Pham [189].](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-25-320.jpg)
![Chapter 2
Lower bounds for linear forms
in complex and p-adic logarithms
We list below several estimates for linear forms in complex and p-adic logarithms, which
will be used throughout the book. We give only few bibliographic references and direct
the reader to the monograph of Waldschmidt [432] for further information, in particular
to its Section 10.4. Most of the results quoted are corollaries of more precise estimates, so
the reader wishing to apply the theory of linear forms in logarithms should better consult
the original papers to find the sharpest bounds available to date.
In this chapter we write completely explicit estimates. This will, however, not be
the case for most of the results presented in the next chapters, where we will often make
no effort to give explicit values for the numerical constants. Throughout, h denotes the
(logarithmic) Weil height; see Definition B.4.
2.1. Lower bounds for linear forms in complex logarithms
We start with a general theorem of Waldschmidt [430,432] on inhomogeneous linear forms
in logarithms of algebraic numbers with algebraic coefficients.
Theorem 2.1. Let n 1 be an integer. Let ˛1; : : : ; ˛n be non-zero algebraic numbers. Let
log ˛1; : : : ; log ˛n be determinations of their logarithm and assume that log ˛1; : : : ; log ˛n
are linearly independent over the rationals. Let ˇ0; : : : ; ˇn be algebraic numbers, not
all zero. Let D be the degree over Q of the number field Q.˛1; : : : ; ˛n; ˇ0; : : : ; ˇn/.
Let E; E
, and A1; : : : ; An be real numbers with
E
E1=D
e1=D
; E
e; E
D
log E
;
and
log Aj max
n
h.˛j /;
E
D
j log ˛j j;
log E
D
o
; 1 j n:
Let B
be a real number with
B
E
; B
max
1jn
D log Aj
log E
; log B
max
0j n
h.ˇj /:](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-27-320.jpg)
![10 2. Lower bounds for linear forms in complex and p-adic logarithms
Then, we have
log jˇ0 C ˇ1 log ˛1 C C ˇn log ˛nj
2nC25
n3nC9
DnC2
log A1 : : : log An log B
log E
.log E/ n 1
:
Assume that we are in the homogeneous rational case, that is, assume that ˇ0 D 0
and ˇ1; : : : ; ˇn are rational integers b1; : : : ; bn with bn ¤ 0. Let B0
be a real number
satisfying
B0
E
; B0
max
1j n 1
n jbnj
log Aj
C
jbj j
log An
o
:
Then, we have
log jb1 log ˛1 C C bn log ˛nj
2nC26
n3nC9
DnC2
log A1 : : : log An log B0
log E
.log E/ n 1
:
In particular, choosing E D e and E
D 3D, we get
log jb1 log ˛1 C C bn log ˛nj
2nC26
n3nC9
DnC2
log.3D/ log A1 : : : log An log B0
:
Proof. This follows from Theorem 9.1 of [432], taking into account Remark 3 on page 303
and Proposition 9.18 of [432].
By Proposition 9.21 of [432], the assumption in Theorem 2.1 that log ˛1; : : : ; log ˛n are
linearly independent over the rationals can be relaxed to the assumptions ˇ0 Cˇ1 log ˛1 C
C ˇn log ˛n ¤ 0 and
D3
.log B0/.log Aj /.log E
/ .log D/.log E/2
; 1 j n;
where B0 D B
in the general case and B0 D B0
in the homogeneous rational case.
Theorem 2.1 plainly includes Theorem 1.5. It also contains Theorem 1.11 and the
lower bound (1.7). Taking n 2 and bn D 1 in the homogeneous rational case of
Theorem 2.1 and setting B WD maxf3; jb1j; : : : ; jbn 1jg, we get
log jb1 log ˛1 C C bn 1 log ˛n 1 C log ˛nj n;D log A1 : : : log An log
B
log An
:
This has many important applications, in particular to Diophantine equations; see Theo-
rems 4.1, 4.3, and 4.5. Assuming that there exists a real number ı such that 0 ı 1
2
and jb1 log ˛1 C C bn 1 log ˛n 1 C log ˛nj e ıB
, we deduce that
B n;D ı 1
log A1 : : : log An 1 log ı 1
log.Dn 1
A1 : : : An 1/
log An:
This (and the discussion below explaining how the linear form (2.1) and the quantity (2.2)
are related) shows that Theorem 2.1 also contains Theorem 1.10.
In the homogeneous rational case, a result similar to Theorem 2.1 was proved in-
dependently by Baker and Wüstholz [37, 38]. Since their estimate (which has a better](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-28-320.jpg)
![2.1. Lower bounds for linear forms in complex logarithms 11
dependence on n than in Theorem 2.1, namely the factor n3n
is replaced by n2n
) does not
include the useful parameters E and B0
, and is superseded by Theorem 2.2 below, we do
not quote it.
The parameter E originates in papers by Shorey [367, 368] and is of interest when
˛1; : : : ; ˛n are real and very close to 1, in which case it can be chosen to be very large.
By assumption, log E cannot exceed D min1j n log Aj . However, in some cases, it can
be taken close to this quantity. To see this in the homogeneous rational case, assume that,
for j D 1; : : : ; n, we have ˛j D 1 C 1
xj
, for an integer xj 3. Then, setting
E D E
D min
1j n
xj and Aj D xj C 1; 1 j n;
we see that, since B0
E, Theorem 2.1 implies the lower bound
log jb1 log ˛1 C Cbn log ˛nj n
log A1 : : : log An
.min1j n log Aj /n 1
log maxf3; jb1j; : : : ; jbnjg:
In the most favourable cases, for example when there exists a real number M such that
max1j n xj .min1j n xj /M
, we get
log jb1 log ˛1 C C bn log ˛nj n Mn 2
max
1j n
log Aj
log maxf3; jb1j; : : : ; jbnjg;
thus replacing the product of the log Aj , as it occurs in the statement of Theorem 2.1, with
their maximum. Further explanations are given below Theorem 2.5, in Chapter 5, and in
Section 10.4.3 of [432].
In the course of this textbook, we mention in passing a few applications of the first
assertion of Theorem 2.1 (at the beginning of Section 3.3 and in Section 3.10), but we
only apply estimates for homogeneous linear forms in logarithms with integer coefficients.
Therefore, we focus our attention on the second assertion of Theorem 2.1 and its subsequent
improvements and refinements.
Let n 2 be an integer. Let ˛1; : : : ; ˛n be non-zero algebraic numbers. Let b1; : : : ; bn
be integers. Let log ˛1; : : : ; log ˛n be any determination of the logarithms of ˛1; : : : ; ˛n.
The theory of linear forms in logarithms provides us with lower bounds for the absolute
value of the linear form
b1 log ˛1 C C bn log ˛n; (2.1)
when it is non-zero. This yields lower bounds for the quantity
j˛b1
1 : : : ˛bn
n 1j; (2.2)
which occurs frequently in Diophantine questions. We choose below to mainly consider
quantities of the form (2.2) and not (2.1). When ˛1; : : : ; ˛n are all real numbers, both
forms are essentially equivalent since log.1 C x/ D x C O.x2
/ in the neighborhood of
the origin. This is however not the case when complex non-real numbers are among
˛1; : : : ; ˛n. Indeed, denoting by log the principal determination of the logarithm, to say
that (2.2) is small does not imply that b1 log ˛1 C C bn log ˛n is close to 0, but merely](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-29-320.jpg)
![12 2. Lower bounds for linear forms in complex and p-adic logarithms
that b1 log ˛1 C C bn log ˛n is close to an integer multiple of 2i. Precisely, we derive
the existence of an integer b0 with jb0j jb1j C C jbnj and such that the linear form
b0 log. 1/ C b1 log ˛1 C C bn log ˛n
is close to 0.
The next statement is a corollary of the, at present time, best known general estimate,
due to Matveev [290, 291]. The crucial improvement on the earlier results [37, 430]
concerns the dependence on the number n of logarithms: it is exponential in n, and not of
the form ncn
as in the earlier estimates.
Theorem 2.2. Let n 1 be an integer. Let ˛1; : : : ; ˛n be non-zero algebraic numbers.
Let D be the degree over Q of a number field containing ˛1; : : : ; ˛n. Let A1; : : : ; An be
real numbers with
log Aj max
n
h.˛j /;
j log ˛j j
D
;
0:16
D
o
; 1 j n:
Let b1; : : : ; bn be integers and set
B D maxfjb1j; : : : ; jbnjg
and
B00
D max
n
1; max
n
jbj j
log Aj
log An
W 1 j n
oo
:
Then, we have
log j˛b1
1 : : : ˛bn
n 1j 330nC4
.nC1/5:5
DnC2
log.eD/ log A1 : : : log An log.enB/
(2.3)
and, if n 2 and ˛1; : : : ; ˛n are all real numbers, we get the better lower bound
log j˛b1
1 : : : ˛bn
n 1j 2 30nC3
n4:5
DnC2
log.eD/ log A1 : : : log An log.eB/:
(2.4)
The same statements hold with B replaced by maxfB00
; nB=.D log An/g in (2.3) and
with B replaced by B00
in (2.4).
Proof that Theorem 2.2 follows from Matveev’s results. Inequality (2.4) with B replaced
or not by B00
is an immediate consequence of Corollary 2.3 of Matveev [291]. Denote
by log the principal determination of the logarithm. If j˛b1
1 : : : ˛bn
n 1j 1
2
, then there
exists an integer b0, with jb0j n B, such that
WD jb0 log. 1/ C b1 log ˛1 C C bn log ˛nj
satisfies j˛b1
1 : : : ˛bn
n 1j
2
. Noticing that j log. 1/j D and h. 1/ D 0, we set
log A0 D
D
and deduce (2.3) from Corollary 2.3 of Matveev [291].
Since
jb0j
log A0
log An
nB
D log An
;
we see that B can be replaced by maxfB00
; nB=.D log An/g in (2.3).](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-30-320.jpg)
![2.1. Lower bounds for linear forms in complex logarithms 13
We stress that the definition of B0
in Theorem 2.1 is slightly different from that of B00
in Theorem 2.2. The lower bound (2.4) with B00
in place of B is crucial for the proof of
Theorem 3.8.
Many Diophantine problems can be reduced to lower bounds for linear forms in two or
three logarithms. While, in the case of three logarithms, we do not have very satisfactory
estimates (but see [298]), Laurent, Mignotte, and Nesterenko [254] obtained in 1995 rather
sharp lower bounds for linear forms in two logarithms. The quality of their result is an
illustration of the method of interpolation determinants, which was introduced in this
context by Laurent [252]. The current best known estimates have been established by
Laurent in 2008 in [253]. We display below an estimate obtained in Corollary 1 of [253]
and three consequences of the main result of [254].
Theorem 2.3. Let ˛1 and ˛2 be multiplicatively independent algebraic numbers. Set
D0
D ŒQ.˛1; ˛2/ W Q=ŒR.˛1; ˛2/ W R. Let A1 and A2 be real numbers such that
log Aj max
n
h.˛j /;
1
D0
;
j log ˛j j
D0
o
; j D 1; 2:
Let b1 and b2 be integers, not both zero, and set
log B0
D max
n
log
jb1j
D0 log A2
C
jb2j
D0 log A1
C 0:21;
20
D0
; 1
o
:
Then, we have the lower bound
log jb1 log ˛1 C b2 log ˛2j 25:2 D04
.log A1/.log A2/.log B0
/2
: (2.5)
For n D 2, the numerical constant in (2.4) exceeds 109
. It has been substantially
reduced in Theorem 2.3. This is crucial for applications to the complete resolution of
Diophantine equations. Very roughly speaking, when a Diophantine problem can be
reduced to linear forms in only two logarithms, then it can (often, this is not always true!
See Problem 13.12) be completely solved.
A weaker version (weaker only in terms of the numerical constants) of Theorem 2.3
is proved in this book; see Theorem 11.1. Note also that, unlike in Theorem 2.3, the
algebraic numbers ˛1 and ˛2 are not assumed to be multiplicatively independent in
Theorem 11.1. This allows us to deduce directly from Theorem 2.3 a slightly weaker
version of Theorem 2.6, see Theorem 11.3.
Observe that the dependence on B0
in Theorem 2.3 is not best possible and worse
than in Theorems 2.1 and 2.2, since log B0
occurs squared. Gouillon [202] established
a lower bound for jb1 log ˛1 C b2 log ˛2j, where the dependence on B0
occurs through a
factor .log B0
/ only. His numerical constants being not so small, it is better for most of
the applications to use the bounds established in [253, 254]. However, to have the best
possible dependence in B0
is essential for the proofs of Theorems 3.3 and 5.1.
As in Theorem 2.1, an extra parameter E was introduced in [254]. We reproduce
below Corollaire 3 of [254].
Theorem 2.4. Let ˛1 and ˛2 be multiplicatively independent positive real algebraic
numbers. Let D be the degree of the number field Q.˛1; ˛2/. Let A1 and A2 be real](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-31-320.jpg)
![14 2. Lower bounds for linear forms in complex and p-adic logarithms
numbers such that
log Aj max
n
h.˛j /;
1
D
;
j log ˛j j
D
o
; j D 1; 2:
Let b1 and b2 be integers, not both zero. Let E be a real number with
E 1 C min
nD log A1
j log ˛1j
;
D log A2
j log ˛2j
o
(2.6)
and set
log B0
D max
n
log
jb1j
D log A2
C
jb2j
D log A1
C log log E C 0:47;
10 log E
D
;
1
2
o
:
Assume furthermore that 2 E minfA3D=2
1 ; A3D=2
2 g. Then,
log jb1 log ˛1 C b2 log ˛2j 35:1D4
.log A1/.log A2/.log B0
/2
.log E/ 3
:
In [254], the authors defined the parameter E to be equal to the right hand side of (2.6).
However, it easily follows from the proof that Theorem 2.4 as stated is correct.
A weaker version (weaker only in terms of the numerical constants) of Theorem 2.4 is
proved in this book; see Theorem 11.2.
The following consequence of Theorem 2.4 emphasizes the rôle of the parameter E in
a particular case which occurs frequently in applications.
Theorem 2.5. Let x1; x2; y1; y2 be positive integers with x1 2, x1 ¤ y1, and
y2 x2 6
5
y2. Let b be a positive integer and assume that .x1
y1
/b
¤ x2
y2
. Define
the parameter by
x2
y2
D 1 C x
2 :
Then, we have log x2 1 and
log
ˇ
ˇ
ˇ
y1
x1
b x2
y2
1
ˇ
ˇ
ˇ
35:2
.log x1/
max
n
1 C
log b
log x2
; 10
o2
: (2.7)
Since x2
y2
1, we always have 1. The estimate is stronger when is very close
to 1, that is, when x2
y2
is very close to 1. Under the assumption of Theorem 2.5, we get
instead of (2.5) a lower bound of the form
.log A1/.log A2/
log.x2
y2
1/
.log B0
/2
;
with A1 D maxfx1; 3g and A2 D maxfx2; 3g. This crucial improvement upon the
“classical” estimate (2.5) turns out to have many spectacular applications, some of which
being given in Chapter 5; see Theorems 5.2, 5.4, and 5.6.
Proof of Theorem 2.5 assuming Theorem 2.4. If y1 x1, then the left hand side of (2.7)
exceeds log.y1
x1
1/, thus it exceeds log x1 and (2.7) holds.
If y1 x1 and x1 5, then .y1
x1
/b
4
5
and (2.7) holds since x2
y2
6
5
.](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-32-320.jpg)
![16 2. Lower bounds for linear forms in complex and p-adic logarithms
The assumptions of Theorem 2.4 are then satisfied and we derive that
log
ˇ
ˇ
ˇlog
x2
y2
b log
x1
y1
ˇ
ˇ
ˇ 35:1 .log x1/
log x2
log E
log B0
log E
2
35:1
.log x1/
log B0
log E
2
;
where
log B0
log E
D max
(
log b
log x2
C 1
log x1
log E
C
log log E C 0:47
log E
; 10
)
max
n
1 C
log b
log x2
; 10
o
:
We conclude by using that every real number z with jzj 1
2
satisfies j log.1 C z/j 2jzj.
This completes the proof of the theorem.
For some applications, we need a lower bound for quantities of the shape j˛b
1j,
where ˛ is a complex algebraic number of modulus 1. Such an estimate does not follow
from Theorem 2.3, since in its statement ˛1 and ˛2 are assumed to be multiplicatively
independent. We display a consequence of Théorème 3 of [254].
Theorem 2.6. Let ˛ be a complex algebraic number of modulus 1 which is not a root of
unity. Let b be a positive integer. Set
D0
D
ŒQ.˛/ W Q
2
; log A D max
n 20
D0
; 11
j log ˛j
D0
C h.˛/
o
;
and
log B0
D max
n 17
D0
;
1
10
p
D0
; log
b
25
C 2:35 C
5:1
D0
o
:
Then, we have
log j˛b
1j 9.D0
/3
.log A/.log B0
/2
:
Proof that Theorem 2.6 follows from Théorème 3 of [254]. Recall that log denotes the
principal determination of the logarithm. If j˛b
1j 1
3
, then there exists an integer b0
with jb0j b such that
j˛b
1j
jb0 log. 1/ C b log ˛j
2
:
We then use the inequalities
1
2D0 log A
C
1
68:9
1
40
C
1
68:9
1
25
to get from Théorème 3 of [254] that
log jb0 log. 1/ C b log ˛j 8:87.D0
/3
.log A/.log B0
/2
:
This yields the desired estimate.](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-34-320.jpg)
![2.2. Multiplicative dependence relations between algebraic numbers 17
2.2. Multiplicative dependence relations between algebraic numbers
The results displayed in the previous section provide us with lower bounds for the quantity
jb1 log ˛1 C C bn log ˛nj;
when it is non-zero. However, in some situations, we derive linear forms in logarithms
that are equal to zero. It is then often useful to find rational integers b0
1; : : : ; b0
n, not all
zero, with small absolute values and such that
b0
1 log ˛1 C C b0
n log ˛n D 0:
A result of this type was given by Loxton and van der Poorten [271]. We quote below a
version of Waldschmidt [432].
Theorem 2.7. Let m 2 be an integer and ˛1; : : : ; ˛m multiplicatively dependent non-
zero algebraic numbers. Let log ˛1; : : : ; log ˛m be any determination of their logarithms.
Let D bethedegreeofthe numberfieldgeneratedby ˛1; : : : ; ˛m over Q. Forj D 1; : : : ; m,
let Aj be a real number satisfying
log Aj max
n
h.˛j /;
j log ˛j j
D
; 1
o
:
Then there exist rational integers n1; : : : ; nm, not all zero, such that
n1 log ˛1 C C nm log ˛m D 0
and
jnj j 11.m 1/D3
m 1 .log A1/ .log Am/
log Aj
; for j D 1; : : : ; m.
A full proof of Theorem 2.7, together with additional references, is given in Chapter 10.
Theorem 2.7 is used in the proof of Theorem 3.16.
2.3. Lower bounds for linear forms in p-adic logarithms
Let p be a prime number and K an algebraic number field. Let OK denote the ring
of integers of K. Let p be a prime ideal in K, lying above the prime number p, and
denote by ep its ramification index, that is, the exponent of p in the decomposition of
the ideal pOK in a product of prime ideals; see Section B.1. For a non-zero algebraic
number ˛ in K, let vp.˛/ denote the exponent of p in the decomposition of the fractional
ideal ˛OK in a product of prime ideals and set
vp.˛/ D
vp.˛/
ep
:
This defines a valuation vp on K which extends the p-adic valuation vp on Q normalized
in such a way that vp.p/ D 1.](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-35-320.jpg)
![18 2. Lower bounds for linear forms in complex and p-adic logarithms
Let ˛1; : : : ; ˛n be elements of K and b1; : : : ; bn non-zero rational integers. We look
for an upper bound for the quantity
vp.˛b1
1 : : : ˛bn
n 1/;
where b1; : : : ; bn denote rational integers, not all zero, when ˛b1
1 : : : ˛bn
n is not equal to 1.
We begin with an easy estimate.
Theorem 2.8. Let p be a prime number, b 2 an integer, and ˛ an algebraic number of
degree D. If ˛b
is not equal to 1, then
vp.˛b
1/
log b
log p
C 2D
pD
1
log p
h.˛/ C 2D
log 2
log p
:
The proof of Theorem 2.8 is given in Section B.3. Theorem 2.8 is trivial unless
vp.˛/ D 0, in which case the factor .pD
1/ corresponds to the upper bound for the
smallest positive integer s such that vp.˛s
1/ is positive. The dependence on p in
Theorem 2.8 was slightly improved by Yamada [442] by means of a subtle variation
of the proof of the main estimate of [129] for linear forms in two p-adic logarithms.
Yamada’s result, reproduced as Theorem 12.3 and established in Chapter 12, has a striking
application to an old conjecture of Erdős; see Theorem 7.11.
The next result is a slight simplification of the estimate given on page 190 of Yu’s
paper [449].
Theorem 2.9. Let p be a prime number and ˛1; : : : ; ˛n algebraic numbers in an algebraic
number field of degree D. Let b1; : : : ; bn denote rational integers such that ˛b1
1 : : : ˛bn
n is
not equal to 1. Let A1; : : : ; An; B be real numbers with
log Aj max
n
h.˛j /;
1
16e2D2
o
; 1 j n;
and
B maxfjb1j; : : : ; jbnj; 3g:
If n 2, then we have
vp.˛b1
1 : : : ˛bn
n 1/.16eD/2.nC1/
n5=2
.log.2nD//2
Dn pD
1
.log p/2
log A1 : : : log An log B:
Theorem 2.9 allows us to improve slightly the dependence on p in Theorem 2.8; see
also Theorem 12.3. Note that (as in Theorems 2.10 and 2.11, but unlike in Theorem 2.12
and in [443]) we do not assume that vp.˛j / D 0 for j D 1; : : : ; n.
The factor .pD
1/ in Theorem 2.9 and in Theorems 2.10 to 2.12 is ultimately due to
the fact that the radius of convergence of the p-adic exponential function is finite, equal
to p 1=.p 1/
. Removing it is a major open problem in the theory. This can be done under
the rather restrictive assumption that vp.˛j 1/ 0 for j D 1; : : : ; n; see [168,169].
Theorem 2.9 should be compared with Theorem 2.2. We have exactly the same de-
pendence on the parameters n, log A1; : : : ; log An, and B. A crucial point in Theorem 2.9
is the dependence on n, which is essential in the proofs of Theorems 8.2 and 8.3 towards](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-36-320.jpg)
![2.3. Lower bounds for linear forms in p-adic logarithms 19
the abc-conjecture. These proofs also require an estimate for vp.˛b1
1 : : : ˛bn
n 1/ with a
(slightly) better dependence on p. For this reason, we quote the result stated on page 30
of [446].
Theorem 2.10. We keep the notation of Theorem 2.9. If n 1, then we have
vp.˛b1
1 : : : ˛bn
n 1/ 12
pD
1
log p
6.n C 1/D
2.nC1/
log.e5
nD/
max
nh.˛1/
log p
; 1
o
max
nh.˛n/
log p
; 1
o
log B:
Yu [449] obtained an improvement of Theorem 2.10, with a better dependence on n,
namely with nn
in place of .n C 1/2n
. We refer to [449] for a precise statement.
The next statement, extracted from page 191 of Yu’s paper [449], is the p-adic analogue
of the main result of [25].
Theorem 2.11. We keep the notation of Theorem 2.9. Let Bn be a real number such that
B Bn jbnj:
Assume that
vp.bn/ vp.bj /; j D 1; : : : ; n:
Let ı be a real number with 0 ı 1
2
. Then, we have
vp.˛b1
1 : : : ˛bn
n 1/ .16eD/2.nC1/
n3=2
.log.2nD//2
Dn pD
1
.log p/2
max
n
.log A1/ .log An/.log T /;
ıB
Bnc0.n; D/
o
; (2.8)
where
T D
Bn
ı
c1.n; D/p.nC1/D
.log A1/ .log An 1/
and
c0.n; D/ D .2D/2nC1
log.2D/ log3
.3D/; c1.n; D/ D 2e.nC1/.6nC5/
D3n
log.2D/:
In particular, we get either
B 2Bn.log A1/ .log An/ c0.n; D/ (2.9)
or
vp.˛b1
1 : : : ˛bn
n 1/ .16eD/2.nC1/
n3=2
.log.2nD//2
Dn pD
.log p/2
.log A1/ .log An/ max
n
1; log
c1.n; D/p.nC1/D
B
c0.n; D/ log An
o
: (2.10)](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-37-320.jpg)
![20 2. Lower bounds for linear forms in complex and p-adic logarithms
The formulation of Theorem 2.11 looks complicated, but it essentially corresponds to
the introduction of the parameter B0
in Theorem 2.1 above. To get the second statement,
we have selected
ı0 D
.log A1/ .log An/Bn
B
c0.n; D/
and distinguished the two cases ı0 1
2
, which gives (2.9), and ı0 1
2
, which, by (2.8)
with ı D ı0, gives (2.10).
In the special case bn D 1, the statements of Theorems 2.2 and 2.11 can be merged;
see Theorem 3.2.8 of [183].
As for linear forms in complex logarithms, the case n D 2 is very important for
applications. The next theorem reproduces one of the corollaries of the main result
of [129].
Theorem 2.12. Let p be a prime number. Let ˛1 and ˛2 be multiplicatively independent
algebraic numbers with vp.˛1/ D vp.˛2/ D 0. Set D D ŒQ.˛1; ˛2/ W Q. Let A1 and A2
be real numbers with
log Aj max
n
h.˛j /;
log p
D
o
; j D 1; 2:
Let b1 and b2 be positive integers and set
log B0
D max
n
log
b1
D log A2
C
b2
D log A1
C log log p C 0:4;
10 log p
D
; 10
o
:
Then, we have the upper bound
vp.˛b1
1 ˛b2
2 /
24p.pD
1/
.p 1/.log p/4
D4
.log A1/.log A2/ .log B0
/2
: (2.11)
The assumption that ˛1 and ˛2 are multiplicatively independent is not really restrictive
for the applications and is useful to get small numerical values in (2.11). The assumption
that ˛1 and ˛2 are p-adic units is not restrictive. Indeed, if only one among vp.˛1/
and vp.˛2/ is positive, then vp.˛b1
1 ˛b2
2 / D 0, while, if both of them are positive,
then vp.˛b1
1 ˛b2
2 / is bounded from below by a positive real number times the minimum
of b1 and b2.
A version of Theorem 2.12 (which is weaker only in terms of the numerical constants)
is proved in this book; see Theorem 12.1.
Unlike Theorem 2.9, Theorem 2.12 cannot be applied to bound vp.˛b
1/ from above
because of the assumption of multiplicative independence, but see Theorem 12.3.
Theorem 2.12 should be compared with Theorem 2.3. In particular, the numerical
constant is very small and the quantity log B0
also occurs squared.
We end this section with an improvement of Theorem 2.12 when ˛1 and ˛2 are rational
numbers which are p-adically close to 1.
Let x1
y1
and x2
y2
be non-zero rational numbers. Theorem 2.13 below, originated in [109],
provides an explicit upper bound for the p-adic valuation of the difference between integer
powers of x1
y1
and x2
y2
.](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-38-320.jpg)
![2.3. Lower bounds for linear forms in p-adic logarithms 21
We suppose that there exist a positive integer g and a real number E with E 1C 1
p 1
and
vp
x1
y1
g
1
E; (2.12)
and at least one of the two following conditions
vp
x2
y2
g
1
E (2.13)
or
vp
x2
y2
g
1
1 and vp.b2/ vp.b1/ (2.14)
is satisfied. For instance, if p is an odd prime number, then (2.12) and (2.13) hold with
g D 1 if x1
y1
and x2
y2
are both congruent to 1 modulo p2
.
We stress that the assumption “(2.13) or (2.14) holds” is more restrictive than the
assumption “vp..x2
y2
/g
1/ 0” made in [109]. However, it seems that the latter
assumption is not sufficient to derive Theorem 2 from Theorem 1 in [109], a fact which
has been overlooked in [109].
Theorem 2.13. Let x1
y1
and x2
y2
be multiplicatively independent rational numbers. Let
b1; b2 be positive integers. Assume that there exist a positive integer g and a real
number E greater than 1 C 1
p 1
such that (2.12) holds, as well as (2.13) or (2.14). Let
A1; A2 be real numbers with
log Aj maxflog jxj j; log jyj j; E log pg; j D 1; 2;
and put
log B0
D max
n
log
b1
D log A2
C
b2
D log A1
C log.E log p/ C 0:4; 6 E log p; 5
o
:
Then, we have the upper bound
vp
x1
y1
b1
x2
y2
b2
36:1 g
E3 .log p/4
.log A1/.log A2/.log B0
/2
; (2.15)
if p is odd or if p D 2 and v2.x2
y2
1/ 2.
Observe that, if x1
y1
and x2
y2
are both congruent to 1 modulo p3
, then we can take E D 3
in Theorem 2.13 and the right hand side of (2.15) can be bounded independently of p.
A weaker version of Theorem 2.13 is proved in this book; see Theorem 12.2.
The parameter E in Theorem 2.13 should be compared with the parameter E in
Theorems 2.1 and 2.4. Let us illustrate its importance with the following example taken
from [117] (see also Exercise 6.11). Apply Theorem 2.13 with the prime number p to
bound from above the p-adic valuation v of .x
y
/b
.1 C cpk
/, for positive integers k, b,
c, x, y, M with k 3, 1 c pMk
, x pk
, and gcd.p; b/ D 1. The latter assumption
implies that, if v 1, then the p-adic valuations of .x
y
/b
1 and x
y
1 are positive](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-39-320.jpg)
![22 2. Lower bounds for linear forms in complex and p-adic logarithms
and equal. Consequently, pk
divides x
y
1 and we apply Theorem 2.13 with g D 1,
A1 D p.MC1/k
, A2 D maxfx; yg, and E D k to get
v
36:1
log p
log A1
E.log p/
.log A2/ .6 C log b/2
;
thus
v
36:1.M C 1/
log p
.log A2/ .6 C log b/2
:
This upper bound is independent of k. It considerably improves the upper bound given
by (2.11) and can be viewed as an analogue of Theorem 2.5 (with 1
replaced with M C1).
2.4. Notes
F Linear forms in one logarithm have been studied by Mignotte and Waldschmidt [300].
Let ˛ be an algebraic number of degree D 2 and of small height. In their lower bound
for j˛ 1j (that is, for j log ˛j), the dependence on D is much better than the one given
by Liouville’s inequality Theorem B.10; see also [134]. Amoroso [7] proved that these
results are essentially sharp; see also [171]. A p-adic analogue has been worked out
in [105].
F In the homogeneous rational case, a complete proof of Theorem 2.1 up to the dependence
on n of the numerical constant has been given by Waldschmidt in [433]. Nesterenko [316]
has written a detailed proof of Theorem 2.2 in the special case where ˛1; : : : ; ˛n are rational
numbers. Aleksentsev [5] established an estimate of a similar strength as Theorem 2.2.
F There is a large gap between the size of the numerical constants in the best known lower
bounds for linear forms in two and in three logarithms. For applications to Diophantine
problems, it would be of greatest interest to improve the lower estimates for linear forms
in three logarithms; see [298] for a step in this direction.
F Bugeaud [111] obtained lower bounds for linear forms in two logarithms and simul-
taneously for several p-adic valuations; see [52,54,111] for applications to Diophantine
equations.
F Simultaneous linear forms in logarithms are discussed in Chapter 9.](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-40-320.jpg)
![3.2. Effective irrationality measures for quotients of logarithms of integers 25
3.2. Effective irrationality measures for quotients
of logarithms of integers
Let a and b be multiplicatively independent positive rational numbers. It follows straight-
forwardly from the first assertion of Theorem 2.1 that the real number log a
log b
is transcendental
and, by arguing as in the proof of Theorem 3.1, we show that it is not a Liouville number
(see Definition A.6) and give an effective upper bound for its irrationality exponent (see
Definition A.3).
Theorem 3.3. Let a1; a2; b1; b2 be positive integers such that the rational numbers a1
a2
and b1
b2
are multiplicatively independent. Then, the effective irrationality exponent of the
real number .log a1
a2
/=.log b1
b2
/ satisfies
eff
log a1
a2
log b1
b2
.log maxfa1; a2g/ .log maxfb1; b2g/:
A strengthening of Theorem 3.3 in the special case where a1
a2
and b1
b2
are rational
numbers close to 1 is given in Section 5.1.
Proof. Let p
q
be a convergent to .log a1
a2
/=.log b1
b2
/ with q 2. Since an irrational
real number and its inverse have the same irrationality exponent, we can assume that
j log a1
a2
j j log b1
b2
j. It then follows from Theorem 2.1 (or from Theorem 2.2) that
log
ˇ
ˇ
ˇ
ˇq log
a1
a2
p log
b1
b2
ˇ
ˇ
ˇ
ˇ .log maxfa1; a2g/ .log maxfb1; b2g/ .log q/:
Thus, there exists an effectively computable, absolute real number C such that
ˇ
ˇ
ˇ
ˇ
log a1
a2
log b1
b2
p
q
ˇ
ˇ
ˇ
ˇ D
jq log a1
a2
p log b1
b2
j
q log b1
b2
q C.log maxfa1;a2g/ .log maxfb1;b2g/
:
This proves the theorem.
The fact that the dependence on B in Theorem 2.2 occurs through the factor .log B/ and
not through the factor .log B/2
, as in Theorem 2.3, is crucial for the proof of Theorem 3.3.
3.3. On the distance between two integral S-units
For a finite set S of prime numbers, an integral S-unit is, by definition, an integer all of
whose prime factors are in S. The following result extends Theorem 3.1. It was proved
by Tijdeman [414] in 1973.
Theorem 3.4. Let S denote a finite, non-empty set of prime numbers and .xj /j 1 the
increasing sequence of all positive integers whose prime factors belong to S. There exists
an effectively computable real number C, depending only on S, such that
xj C1 xj xj .log 2xj / C
; j 1:](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-43-320.jpg)
![26 3. First applications
Proof. Write S D fq1; : : : ; qsg, where q1; : : : ; qs are prime numbers and q1 qs.
Let u; v with 3 u v be two consecutive elements in the sequence .xj /j 1 and write
u D
s
Y
iD1
qui
i ; v D
s
Y
iD1
qvi
i :
Without any loss of generality, we can assume that s 2, v 2u, and that u and v are
coprime. Thus, for every i D 1; : : : ; s, at least one of ui ; vi is zero. Set
ƒ WD
s
Y
iD1
qvi ui
i 1 D
v
u
1:
Put B D maxf3; jv1 u1j; : : : ; jvs usjg and observe that
2B=3
qB=3
1 v 2u: (3.4)
It follows from Theorem 2.2 that
log ƒ 30sC4
s5
s
Y
iD1
log qi
.log eB/:
Combined with (3.4), this gives
log
v
u
1
30sC4
s5
s
Y
iD1
log qi
log
3e log.2u/
log 2
:
Taking the exponential of both sides and multiplying by u, we get the assertion of the
theorem.
In view of the next result, established by Tijdeman [415] in 1974, Theorem 3.4 is
essentially best possible.
Theorem 3.5. Let P be an integer and S a finite set of at least two distinct prime numbers
all being at most equal to P . Let .xj /j 1 be the increasing sequence of all positive
integers whose prime divisors belong to S. There exists an effectively computable real
number C, depending only on P , such that
xj C1 xj xj .log 2xj / C
; j 1:
It is an open problem of Erdős to establish that, for S being the set of all prime numbers
less than P , Theorem 3.5 holds with a real number C D C.P / which tends to infinity
with P .
3.4. Effective irrationality measures for n-th roots
of algebraic numbers
We have seen in Theorem 3.3 how the theory of linear forms in logarithms can be applied
to give an effective upper bound for the irrationality exponent of certain transcendental
real numbers. The next result addresses a special class of real algebraic numbers.](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-44-320.jpg)
![3.5. On the greatest prime factor of values of integer polynomials 27
Theorem 3.6. Let n 3 be an integer and a positive, real algebraic number. Then, the
effective irrationality exponent of n
p
satisfies
eff. n
p
/ log n:
Furthermore, if a; b are integers with 1 b a, then we have
eff. n
p
a=b / .log a/.log n/;
and, if a 2n
, then we get
eff. n
p
a=b / 11000 log a: (3.5)
When n is sufficiently large in terms of , Theorem 3.6 improves the upper bound
eff. n
p
/ n deg./, given by Liouville’s Theorem A.5. It remains, however, very
far from Roth’s Theorem A.7, which asserts that . n
p
/ D 2 but gives no information
on eff. n
p
/.
In Section 5.2 we considerably strengthen Theorem 3.6 when is a rational number
very close to 1.
Proof. Let n 3 be an integer and a positive, real algebraic number. Let p
q
be a
convergent of n
p
with q 2. Observe that
ˇ
ˇ
ˇ
n
p
p
q
ˇ
ˇ
ˇ
1
2n maxf1; jjg
ˇ
ˇ
ˇ
p
q
nˇ
ˇ
ˇ:
It directly follows from Theorem 2.2 that
log
ˇ
ˇ
ˇ
p
q
nˇ
ˇ
ˇ C./.log q/.log n/;
where C./ is an effectively computable real number which can be expressed in terms of
the height and the degree of . If is the rational number a
b
with 1 b a, then C./ can
be taken to be an absolute real number times log a. This proves the first two statements of
the theorem.
For large values of n, we obtain a stronger result (not only numerically) by applying
Theorem 2.3. Indeed, if a; b are integers with 1 b a, it directly gives the lower bound
log
ˇ
ˇ
ˇn log
p
q
log
a
b
ˇ
ˇ
ˇ 25:2.log p/.log a/
max
n
log
2n
log a
C 0:21; 20
o2
;
if p is sufficiently large. From this and under the assumption a 2n
, we deduce (3.5).
3.5. On the greatest prime factor of values of integer polynomials
Let f .X/ be a non-zero integer polynomial with at least two distinct roots. Let n be an
integer. Using the theory of linear forms in logarithms, Shorey and Tijdeman [374] were
the first to give an effective lower bound for the greatest prime factor of f .n/ as n tends](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-45-320.jpg)
![28 3. First applications
to infinity. In 1998 Tijdeman [418] noticed that, for the polynomial X.X C 1/, the use of
Theorem 2.2 allows us to improve the results of [374]. In his doctoral thesis, Haristoy [221]
extended Tijdeman’s result to an arbitrary polynomial f .X/. For an integer n, let denote
by P Œn its greatest prime factor with the convention that P Œ0 D P Œ˙1 D 1.
Theorem 3.7. Let f .X/ be an integer polynomial with at least two distinct roots. Then,
we have
P Œf .n/ f log log n
log log log n
log log log log n
; for n 107
: (3.6)
A result similar to Theorem 3.7 has been established in [217]; see also Chapter 8.
If we apply Theorem 2.1 (with a weaker dependence on the number of logarithms in
the linear form than in Theorem 2.2) instead of Theorem 2.2, then we would get the weaker
lower bound
P Œf .n/ f log log n; for n 100;
which was obtained in [374].
Proof. To avoid technical complications, we treat only the case of the polynomial X.X C1/.
For an integer n at least equal to 100, write
n.n C 1/ D pu1
1 : : : puk
k
;
where p1 p2 denotes the sequence of prime numbers in increasing order and the
integers u1; : : : ; uk are non-negative. There exist disjoint non-empty subsets I and J of
f1; : : : ; kg such that I [ J D f1; : : : ; kg and
n D
Y
i2I
pui
i ; n C 1 D
Y
j 2J
p
uj
j :
Then, there exist 1; : : : ; k in f˙1g such that
n C 1
n
1 D jp1u1
1 : : : pk uk
k
1j D
1
n
: (3.7)
Observe that
n C 1 2uj
; j D 1; : : : ; k;
thus
max
1j k
uj 2 log n: (3.8)
Since pj j log j , for j D 2; : : : ; k, by Theorem D.2, we deduce from (3.8) and
Theorems 2.2 and D.2 that there exist effectively computable, absolute real numbers
c1; c2; c3 such that
log jp1u1
1 : : : pk uk
k
1j ck
1 .log p1/ .log pk/ .log log n/
c
k log log k
2 .log log n/;
thus, by (3.7),
log n c
k log log k
3 log log n :](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-46-320.jpg)
![3.5. On the greatest prime factor of values of integer polynomials 29
By Theorem D.2, this implies that
pk k log k log log n
log log log n
log log log log n
;
as asserted.
The general case of an arbitrary monic, integer polynomial with distinct roots ˛ and ˇ
rests on the same idea, but is technically more complicated. Let OK denote the ring
of integers of the algebraic number field K WD Q.˛; ˇ/ and h its class number. Let n
be an integer greater than j˛j and jˇj. Write the integer ideal .n ˛/.n ˇ/OK as a
product of powers of prime ideals and raise this equality to the power h. Then, the quantity
.n ˛/h
.n ˇ/ h
, which is equal to 1 C O.1
n
/, can, with the help of Proposition C.5, be
expressed as a product of powers of algebraic numbers in K whose heights are controlled.
It then follows from Theorem 2.2 that this product cannot be too close to 1, hence the
result. We leave the details to the reader.
Let S be a finite non-empty set of prime numbers. For a non-zero integer n, write
ŒnS WD
Y
p2S
jnj 1
p ;
where j jp is the p-adic absolute value normalized such that jpj 1
p D p. Said differently,
ŒnS is the greatest divisor of n, all of whose prime factors belong to S.
In 1984, Stewart [396] applied the theory of linear forms in logarithms to prove non-
trivial effective upper bounds for Œn.n C 1/ : : : .n C k/S , for any positive integer k. His
result was later extended by Gross and Vincent [205] as follows; see also [122].
Theorem 3.8. Let f .X/ be an integer polynomial with at least two distinct roots and S
a finite, non-empty set of prime numbers. There exist effectively computable positive real
numbers c1 and c2, depending only on f .X/ and S, such that, for every non-zero integer n
which is not a root of f .X/, we have
Œf .n/S c1jf .n/j1 c2
:
Proof. To avoid technical complications, we treat only the case of the polynomial X.X C1/.
We leave to the reader the details of the general case. Let n be an integer with jnj 2.
Write S D fq1; : : : ; qsg, where s is the cardinality of S and q1 qs. Let u1; : : : ; us
be non-negative integers and a an integer coprime with q1 qs such that
n.n C 1/ D qu1
1 : : : qus
s a:
There exist disjoint non-empty subsets I and J of f1; : : : ; sg such that I [J D f1; : : : ; sg
and integers a0
; a00
such that a D a0
a00
and
n D a0
Y
i2I
qui
i ; n C 1 D a00
Y
j2J
q
uj
j :
Then, there exist 1; : : : ; s in f˙1g such that
1
n
D
n C 1
n
1 D q1u1
1 : : : qsus
s
a00
a0
1 DW ƒ:](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-47-320.jpg)
![3.6. On the greatest prime factor of terms of linear recurrence sequences 31
factors as
G.z/ D
t
Y
iD1
.z ˛i /`i
;
where ˛1; : : : ; ˛t are distinct algebraic numbers, ordered in such a way that j˛1j
j˛t j, and `1; : : : ; `t are positive integers. Then, (see e.g. [179] or Chapter C of [376])
there exist polynomials f1.X/; : : : ; ft .X/ of degrees less than `1; : : : ; `t , respectively,
and with coefficients in Q.˛1; : : : ; ˛t /, such that
un D f1.n/˛n
1 C C ft .n/˛n
t ; for n 0:
The recurrence sequence .un/n0 is said to be degenerate if there are integers i; j with
1 i j t such that ˛i
˛j
is a root of unity. We say that .un/n0 has a dominant root
if j˛1j j˛2j and f1.X/ is not the zero polynomial.
The next result was established in [399]; see also [395].
Theorem 3.9. Let u WD .un/n0 be a recurrence sequence of integers given by
un D f1.n/˛n
1 C C ft .n/˛n
t ; for n 0;
and having a dominant root ˛1. For any integer n greater than 100 and such that un is
not equal to f1.n/˛n
1 , the greatest prime factor of un satisfies
P Œun u log n
log log n
log log log n
: (3.9)
Proof. We establish a slightly more general result. We consider a sequence of non-zero
integers v WD .vn/n0 with the property that there are in .0; 1/ and a positive real
number C such that
jvn f .n/˛n
j C j˛jn
; n 0;
where f .X/ is a non-zero polynomial whose coefficients are algebraic numbers and ˛
is an algebraic number with j˛j 1. Clearly, a recurrence sequence having a dominant
root ˛ has the above property.
Let p1; p2; : : : be the sequence of prime numbers in increasing order. Let n be a
positive integer such that f .n/ is non-zero and vn ¤ f .n/˛n
. Assume that the greatest
prime factor of vn is equal to pk, that is, assume that
vn D pr1
1 prk
k
;
where r1; : : : ; rk are non-negative integers and rk 1. Then, we have
ƒ WD jpr1
1 prk
k
f .n/ 1
˛ n
1j Cjf .n/j 1
j˛j. 1/n
;
and, since 1, we get
log ƒ v . n/:
As vn ¤ f .n/˛n
, the quantity ƒ is non-zero and Theorem 2.2 implies that
log ƒ v ck
1 .log p1/ .log pk/ .log n/2
;](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-49-320.jpg)
![32 3. First applications
where c1 is an effectively computable, positive, absolute real number. Comparing both
estimates for log ƒ, we obtain that
n v .c1 log pk/k
.log n/2
;
thus, by Theorem D.2,
log n v
pk
log pk
log log pk:
This implies that
P Œvn D pk v log n
log log n
log log log n
;
and the theorem is established.
The lower bound for P Œun in (3.9) is of similar strength as the one for P Œf .n/ in
Theorem 3.7, since junj grows exponentially fast in n. Both proofs show that f in (3.6)
and u in (3.9) can be replaced by .1 /, for any positive and any sufficiently large
integer n.
3.7. Perfect powers in linear recurrence sequences
We continue the study of arithmetical properties of linear recurrence sequences of integers
having a dominant root and show that no term of such a sequence can be a very large
power of an integer greater than one.
Theorem 3.10. Let u WD .un/n0 be a recurrence sequence of integers given by
un D f1.n/˛n
1 C C ft .n/˛n
t ; for n 0;
and having a dominant root ˛1. Assume that ˛1 is a simple root, that is, the polynomial
f1.X/ is equal to a non-zero algebraic number f1. Then, the equation
un D yq
;
in integers n; y; q with q a prime number, jyj 2, and un ¤ f1˛n
1 , implies that
q u 1:
Theorem 3.10 was proved in [372]; see also [190,325]. More can be said in the case
of binary recurrence sequences, see Section 6.9.
Proof. We establish a slightly more general result. We consider a sequence of non-zero
integers v WD .vn/n0 with the property that there are in .0; 1/ and a positive real
number C such that
jvn f ˛n
j C j˛jn
; n 0;
where f is a non-zero algebraic number and ˛ is an algebraic number with j˛j 1.
Let n; y; q with q a prime number be such that vn D yq
, jyj 2, and vn ¤ f ˛n
.
Then, we have
jf 1
˛ n
yq
1j
C
jf j
j˛j. 1/n
: (3.10)](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-50-320.jpg)
![3.7. Perfect powers in linear recurrence sequences 33
There exist integers k and r such that n D kq C r with jrj q
2
. Rewriting (3.10) as
ƒ WD
ˇ
ˇ
ˇf 1
y
˛k
q
˛ r
1
ˇ
ˇ
ˇ
C
jf j
j˛j. 1/n
;
we observe that
log ƒ v . n/: (3.11)
By Theorem B.5, the height of ˛k
y
is bounded from above by the sum of log jyj and k
times the height of ˛. Since
q log jyj v n .k C 1/q and
kq
2
n v q log jyj; (3.12)
we get that h.˛k
=y/ v log jyj. It then follows from Theorem 2.2 that
log ƒ v .log jyj/.log q/;
which, combined with (3.11) and (3.12), gives
q log jyj v n v .log jyj/.log q/;
and we get q v 1. This proves the theorem.
We point out a very special case of Theorem 3.10.
Corollary 3.11. Let ˛ and ˇ be algebraic integers such that ˛ C ˇ, ˛ˇ are rational
integers and ˛ jˇj. Then, the equation
˛n
ˇn
˛ ˇ
D yq
.resp., ˛n
C ˇn
D yq
/; (3.13)
in integers n; y; q with jyj 2 and q a prime number, implies that
q ˛;ˇ 1:
Let us discuss a celebrated example of resolution of equations of the form (3.13),
namely the determination of all the perfect powers among the Fibonacci and Lucas num-
bers. Set D 1C
p
5
2
and observe that 1
D 1
p
5
2
. Let the integer sequences .Fn/n0
and .Ln/n0 be defined respectively by
Fn D
n
. 1
/n
. 1/
; Ln D n
C . 1
/n
;
so that the first elements of the Fibonacci sequence .Fn/n0 and of the Lucas sequence
.Ln/n0 are given by
0; 1; 1; 2; 3; 5; 8; 13; 21; 34; 55; 89; 144; 233; : : :
and
2; 1; 3; 4; 7; 11; 18; 29; 47; 76; : : : ;
respectively. The Fibonacci sequence is an emblematic example of a Lucas–Lehmer
sequence (see Section 7.2). The following result has been established in [137].](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-51-320.jpg)
![4. Barium Ba = 68.5
5. Bismuth Bi = 213
6. Cadmium Cd = 56
7. Calcium Ca = 20
8. Cerium Ce = 47
9. Chromium Cr = 26.7
10. Cobalt Co = 29.5
11. Copper Cu = 31.7
12. Donarium
13. Didymium D
14. Erbium E
15. Gold Au = 197
16. Glucinum Gl
17. Iron Fe = 28
18. Ilmenium Il
19. Iridium Ir = 99
20. Lead Pb = 103.7
21. Lanthanium La
22. Lithium Li = 6.5
23. Magnesium Mg = 12.2
24. Manganese Mn = 27.6
25. Mercury Hg = 100
26. Molybdenum Mo = 46
27. Nickel Ni = 29.6
28. Norium
29. Niobium Nb
30. Osmium Os = 99.6
31. Platinum Pt = 98.7
32. Potassium K = 39.2
33. Palladium Pd = 53.3
34. Pelopium Pe
35. Rhodium R = 52.2
36. Rhuthenium Ru = 52.2
37. Silver Ag = 108.1
38. Sodium Na = 23
39. Strontium Sr = 43.8
40. Tin Sn = 59
41. Tantalum Ta = 184
42. Tellurium Te = 64.2
43. Terbium Tb
44. Thorium Th = 59.6
45. Titanium Ti = 25
46. Tungsten W[A]= 95
47. Uranium U = 60
48. Vanadium V = 68.6](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-75-320.jpg)
![49. Yttrium Y
50. Zinc Zn = 32.6
51. Zirconium Zr = 22.4
(N.B. The elements printed in italics are at present unimportant.)
[A] From the mineral Wolfram, and now exceedingly valuable, as when alloyed with iron it is
harder than, and will bore through steel.
A few words will suffice to explain the meaning of the terms which head the names, letters, and
numbers of the Table of Elements. The names of the elements have very interesting derivations,
which it is not the object of this work to go into; the symbols are abbreviations, ciphers of the
simplest kind, to save time and trouble in the frequent repetition of long words, just as the signs +
plus, and - minus, are used in algebraic formulæ. For instance—the constant recurrence of water in
chemical combinations must be named, and would involve the most tedious repetition; water consists
of oxygen and hydrogen, and by taking the first letter of each word we have an instructive symbol,
which not only gives us an abbreviated term for water, but also imparts at once a knowledge of its
composition by the use of the letters, HO.
Again, to take a more complex example, such as would occur in the study of organic chemistry—a
sentence such as the hydrated oxide of acetule, is written at once by C4H4O2, the figures referring to
the number of equivalents of each element—viz., 4 equivalents of C, the symbol for carbon, 4 of H
(hydrogen), and 2 of O (oxygen).
The long word paranaphthaline, a substance contained in coal tar, is disposed of at once with the
symbols and figures C30H12.
The figures in the third column are, however, the most interesting to the precise and mathematically
exact chemist. They represent the united labours of the most painstaking and learned chemists, and
are the exact quantities in which the various elements unite. To quote one example: if 8 parts by
weight of oxygen—viz., the combining proportions of that element—are united with 1 part by weight
of hydrogen, also its combining number, the result will be 9 parts by weight of water; but if 8 parts of
oxygen and 2 parts of hydrogen were used, one only of the latter could unite with the former, and the
result would be the formation again of 9 parts of water, with an overplus of 1 equivalent of hydrogen.
It is useless to multiply examples, and it is sufficient to know that with this table of numbers the
figures of analysis are obtained. Supposing a substance contained 27 parts of water, and the oxygen
in this had to be determined, the rule of proportion would give it at once, 9: 27:: 8: 24. 9 parts of
water are to 27 parts as 8 of oxygen (the quantity contained in 9 parts of water) are to the answer
required—viz., 24 of oxygen. The names, symbols, and combining proportions being understood, we
may now proceed with the performance of many interesting
CHEMICAL EXPERIMENTS.
As the permanent gases head the list, they will first engage our attention, beginning with the element
oxygen—Symbol O, combining proportion 8. There is nothing can give a better idea of the enormous
quantity of oxygen present in the animal, vegetable, and mineral kingdoms, than the statement that it
represents one-third of the weight of the whole crust of the globe. Silica, or flint, contains about half
its weight of oxygen; lime contains forty per cent.; alumina about thirty-three per cent. In these
substances the element oxygen remains inactive and powerless, chained by the strong fetters of
chemical affinity to the silicium of the flint, the calcium of the lime, and the aluminum of the alumina.
If these substances are heated by themselves they will not yield up the large quantity of oxygen they
contain.](https://image.slidesharecdn.com/3425091-250620041307-e213e8ef/85/Linear-Forms-In-Logarithms-And-Applications-Yann-Bugeaud-76-320.jpg)
Linear Forms In Logarithms And Applications Yann Bugeaud
- 1. Linear Forms In Logarithms And Applications Yann Bugeaud download https://ebookbell.com/product/linear-forms-in-logarithms-and- applications-yann-bugeaud-6850182 Explore and download more ebooks at ebookbell.com
- 2. Bugeaud IRMA 28 | FONT: Rotis Sans | Farben: Pantone 287, Pantone 116 | 170 x 240 mm | RB: 15??? mm Yann Bugeaud Linear Forms in Logarithms and Applications The aim of this book is to serve as an introductory text to the theory of linear forms in the logarithms of algebraic numbers, with a special emphasis on a large variety of its applications. We wish to help students and researchers to learn what is hidden inside the blackbox ‘Baker’s theory of linear forms in logarithms’ (in complex or in p-adic logarithms) and how this theory applies to many Diophantine problems, including the effective resolution of Diophantine equations, the abc-conjecture, and upper bounds for the irrationality measure of some real numbers. Written for a broad audience, this accessible and self-contained book can be used for graduate courses (some 30 exercises are supplied). Specialists will appreciate the inclusion of over 30 open problems and the rich bibliography of over 450 references. ISBN 978-3-03719-183-5 www.ems-ph.org Yann Bugeaud Linear Forms in Logarithms and Applications Yann Bugeaud Linear Forms in Logarithms and Applications IRMA Lectures in Mathematics and Theoretical Physics 28
- 3. IRMA Lectures in Mathematics and Theoretical Physics 28 Edited by Christian Kassel and Vladimir G. Turaev Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 rue René Descartes 67084 Strasbourg Cedex France
- 4. IRMA Lectures in Mathematics and Theoretical Physics Edited by Christian Kassel and Vladimir G. Turaev This series is devoted to the publication of research monographs, lecture notes, and other material arising from programs of the Institut de Recherche Mathématique Avancée (Strasbourg, France). The goal is to promote recent advances in mathematics and theoretical physics and to make them accessible to wide circles of mathematicians, physicists, and students of these disciplines. For a complete listing see our homepage at www.ems-ph.org. 8 AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries, Oliver Biquard (Ed.) 9 Differential Equations and Quantum Groups, D. Bertrand, B. Enriquez, C. Mitschi, C. Sabbah and R. Schäfke (Eds.) 10 Physics and Number Theory, Louise Nyssen (Ed.) 11 Handbook of Teichmüller Theory, Volume I, Athanase Papadopoulos (Ed.) 12 Quantum Groups, Benjamin Enriquez (Ed.) 13 Handbook of Teichmüller Theory, Volume II, Athanase Papadopoulos (Ed.) 14 Michel Weber, Dynamical Systems and Processes 15 Renormalization and Galois Theory, Alain Connes, Frédéric Fauvet and Jean-Pierre Ramis (Eds.) 16 Handbook of Pseudo-Riemannian Geometry and Supersymmetry, Vicente Cortés (Ed.) 17 Handbook of Teichmüller Theory, Volume III, Athanase Papadopoulos (Ed.) 18 Strasbourg Master Class on Geometry, Athanase Papadopoulos (Ed.) 19 Handbook of Teichmüller Theory, Volume IV, Athanase Papadopoulos (Ed.) 20 Singularities in Geometry and Topology. Strasbourg 2009, Vincent Blanlœil and Toru Ohmoto (Eds.) 21 Faà di Bruno Hopf Algebras, Dyson–Schwinger Equations, and Lie–Butcher Series, Kurusch Ebrahimi-Fard and Frédéric Fauvet (Eds.) 22 Handbook of Hilbert Geometry, Athanase Papadopoulos and Marc Troyanov (Eds.) 23 Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics, Lizhen Ji and Athanase Papadopoulos (Eds.) 24 Free Loop Spaces in Geometry and Topology, Janko Latschev and Alexandru Oancea (Eds.) 25 Takashi Shioya, Metric Measure Geometry. Gromov‘s Theory of Convergence and Concentration of Metrics and Measures 26 Handbook of Teichmüller Theory, Volume V, Athanase Papadopoulos (Ed.) 27 Handbook of Teichmüller Theory, Volume VI, Athanase Papadopoulos (Ed.)
- 5. Yann Bugeaud Linear Forms in Logarithms and Applications
- 6. Author: Yann Bugeaud Université de Strasbourg, CNRS Institut de Recherche Mathématique Avancée, UMR 7501 7, rue René Descartes 67084 Strasbourg CEDEX France E-mail: bugeaud@math.unistra.fr 2010 Mathematics Subject Classification: Primary: 11-02, 11J86, 11D; Secondary: 11B37, 11D25, 11D41, 11D59, 11D61, 11D75, 11D88, 11J25, 11J81, 11J82 Key words: Baker’s theory, linear form in logarithms, Diophantine equation, Thue equation, abc-conjecture, primitive divisor, irrationality measure, p-adic analysis ISBN 978-3-03719-183-5 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2018 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A21 CH-8092 Zürich Switzerland Email: info@ems-ph.org Homepage: www.ems-ph.org Typeset using the author’s TEX files: Nicole Bloye, Cardiff, UK Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1
- 7. Preface Hay que saber buscar aunque uno no sepa qué es lo que busca. Roberto Bolaño — Les maths, m’sieur, ça fonctionne toujours à côté de ses grolles. Jean Amila In 1748 Leonhard Euler published Introductio in analysin infinitorum where, among several fundamental results, he established the relationship ei D 1 and gave explicitly the continued fraction expansions of e and e2 . He also made a conjecture concerning the nature of quotients of logarithms of rational numbers, which can be formulated as follows: For any two positive rational numbers r, s with r different from 1, the number log s=log r is either rational (in which case there are non-zero integers a, b such that ra D sb ) or transcendental. Recall that a complex number is called algebraic if it is a root of a non-zero polynomial with integer coefficients and a complex number which is not algebraic is called transcen- dental. Euler’s conjecture implies, for example, that 2 p 2 is irrational (if it were rational, then log 2 p 2 divided by log 2, which is equal to p 2, would be rational or transcendental). It can be reformulated as follows: If a is a positive rational number different from 1 and ˇ an irrational real algebraic number, then aˇ is irrational. In 1900, David Hilbert proposed a list of twenty-three open problems and presented ten of them in Paris at the second International Congress of Mathematicians. His seventh problem expands the arithmetical nature of the numbers under consideration in Euler’s conjecture and asks whether (observe that e D . 1/ i ) the expression ˛ˇ for an algebraic base ˛ different from 0 and 1 and an irrational algebraic exponent ˇ, e.g. the number 2 p 2 or e , always represents a transcendental or at least an irrational number. Here and below, unless otherwise specified, by algebraic number we mean complex algebraic number. Hilbert believed that the Riemann Hypothesis would be settled long before his seventh problem. This was not the case: the seventh problem was eventually
- 8. vi Preface solved in 1934, independently and simultaneously, by Aleksandr Gelfond and Theodor Schneider, by different methods. They established that, for any non-zero algebraic numbers ˛1; ˛2; ˇ1; ˇ2 with log ˛1 and log ˛2 linearly independent over the rationals (here and below, log denotes the principal determination of the logarithm function), we have ƒ2 WD ˇ1 log ˛1 C ˇ2 log ˛2 6D 0: Since the formulation is different, let us add some explanation. Under the hypotheses of Hilbert’s seventh problem, the complex numbers log ˛ and log ˛ˇ are linearly independent over the rationals and, assuming furthermore that ˛ˇ is algebraic, we derive from the Gelfond–Schneider theorem that ˇ, equal to the quotient of the logarithm of ˛ˇ by the logarithm of ˛, cannot be algebraic, a contradiction. Subsequently, Gelfond derived a lower bound for jƒ2j and, a few years later, he realized that an extension of his result to linear forms in an arbitrarily large number of logarithms of algebraic numbers would enable one to solve many challenging problems in Diophantine approximation and in the theory of Diophantine equations. This program was realized by Alan Baker in a series of four papers published between 1966 and 1968 in the journal Mathematika. He made the long awaited breakthrough, by showing that, if ˛1; : : : ; ˛n are non-zero algebraic numbers such that log ˛1; : : : ; log ˛n are linearly independent over the rationals, and if ˇ1; : : : ; ˇn are non-zero algebraic numbers, then ƒn WD ˇ1 log ˛1 C C ˇn log ˛n 6D 0: In addition, he derived a lower bound for jƒnj, thereby giving the expected extension of the Gelfond–Schneider theorem. In his work, Baker generated a large class of transcendental numbers not previously identified and showed how the underlying theory can be used to answer a wide range of Diophantine problems, including the effective resolution of many classical Diophantine equations. He was awarded a Fields Medal in 1970 at the International Congress of Mathematicians in Nice. It then became clear that further progress, refinements, and extensions of the theory would have important consequences. This area was at that time flourishing and developing very rapidly, both from a theoretical point of view (with improvements obtained by Baker and Feldman, among others, on the lower bounds for jƒnj) and regarding its applications. A spectacular achievement was the proof by Robert Tijdeman in 1976 that the Catalan equation xm yn D 1, in the integer unknowns x; y; m; n all greater than 1, has only finitely many solutions (Preda Mihăilescu established in 2002 that 32 23 D 1 is the only solution to this equation). The aim of the present monograph is to serve as an introductory text to Baker’s theory of linear forms in the logarithms of algebraic numbers, with a special emphasis on a large variety of its applications, mainly to Diophantine questions. We wish to help students and researchers to learn what is hidden inside the blackbox “Baker’s theory of linear forms in logarithms” (in complex or in p-adic logarithms) and how this theory applies to many Diophantine problems. Chapter 1 gives the reader a concise historical introduction to the theory. In Chapter 2, we gather several explicit lower bounds for jƒnj and its p-adic analogue, which were es- tablished by Waldschmidt, Matveev, Laurent, Mignotte and Nesterenko, Yu, and Bugeaud
- 9. Preface vii and Laurent, and which will be used in the subsequent chapters. In all but one of these estimates, ˇ1; : : : ; ˇn are integers, a special case sufficient for most of the applications. The lower bounds are then expressed in terms of the maximum B of their absolute values and take the form log jƒnj c.n; D/ .log 2A1/ : : : .log 2An/ .log 2B/; where c.n; D/ is an explicit real number depending only on n and the degree D of the algebraic number field generated by ˛1; : : : ; ˛n and Aj is the maximum of the absolute values of the coefficients of the minimal defining polynomial of ˛j over the rational integers, for j D 1; : : : ; n. The crucial achievements of Baker are the logarithmic dependence on B and the fact that an admissible value for c.n; D/ can be explicitly computed. We consider in Chapter 3 Diophantine problems for which the reduction to linear forms in complex logarithms is almost straightforward. These problems include explicit lower bounds for the distance between powers of 2 and powers of 3, effective irrationality measures for n-th roots of rational numbers, lower bounds for the greatest prime factor of n.n C 1/, where n is a positive integer, perfect powers in linear recurrence sequences of integers, etc. Chapter 4 is devoted to applications to classical families of Diophantine equations. In the works of Thue and Siegel, it was established that unit equations, Thue equations, and super- and hyperelliptic equations have only finitely many integer solutions, but the proofs were ineffective, in the sense that they did not yield upper bounds for the absolute values of the solutions and, consequently, were of very little help for the complete resolution of the equations. The theory of linear forms in logarithms induced dramatic changes in the field of Diophantine equations and we explain how it can be applied to establish, in an effective way, that unit equations, Thue equations, super- and hyperelliptic equations, the Catalan equation, etc., have only finitely many integer solutions. This chapter also contains a complete proof, following Bilu and Bugeaud [72], of an effective improvement of Liouville’s inequality (which states that an algebraic number of degree d cannot be approximated by rational numbers at an order greater than d) derived ultimately from an estimate for linear forms in two complex logarithms proved in Chapter 11. When the algebraic numbers ˛1; : : : ; ˛n occurring in the linear form ƒn are all rational numbers very close to 1, the lower bounds for jƒnj can be considerably improved. Several applications of this refinement are listed in Chapter 5. They include effective irrationality measures for n-th roots of rational numbers close to 1 and striking results on the Thue equation axn byn D c. Chapter 6 presents various applications of the theory of linear forms in p-adic log- arithms, in particular towards Waring’s problem and, again, to perfect powers in linear recurrence sequences of integers. It also includes extensions of results established in Chapter 4: unit equations, Thue equations, super- and hyperelliptic equations have only finitely many solutions in the rational numbers, whose denominators are divisible by prime numbers from a given, finite set, and, moreover, the size of these solutions can be effectively bounded. Primitive divisors of terms of binary recurrence sequences are discussed in Chapter 7. We partially prove a deep result of Bilu, Hanrot, and Voutier [77] on the primitive
- 10. viii Preface divisors of Lucas and Lehmer numbers and discuss some of its applications to Diophantine equations. Then, following Stewart [400], we confirm a conjecture of Erdős and show that, for every integer n 3, the greatest prime factor of 2n 1 exceeds some positive real number times n p log n= log log n. In Chapter 8, we follow Stewart and Yu [405] to establish partial results towards the abc-conjecture, which claims that, for every positive real number , there exists a positive real number ./, depending only on , such that, for all coprime, positive integers a; b; and c with a C b D c, we have c ./ Y pjabc p 1C ; the product being taken over the distinct prime factors of abc. Specifically, we show how to combine complex and p-adic estimates to prove the existence of an effectively computable positive real number such that, for all positive coprime integers a; b; and c with a C b D c, we have log c Y pjabc p 1=3 log Y pjabc p 3 : There are only a few known applications of the theory of simultaneous linear forms in logarithms, developed by Loxton in 1986. Two of them are presented in Chapter 9. A first gives us an upper bound for the number of perfect powers in the interval ŒN; N C p N, for every sufficiently large integer N. A second shows that, under a suitable assumption, a system of two Pellian equations has at most one solution. Given a finite set of multiplicatively dependent algebraic numbers, we establish in Chapter 10 that these numbers satisfy a multiplicative dependence relation with small exponents. A key ingredient for the proof is a lower bound for the Weil height of a non-zero algebraic number which is not a root of unity. Full proofs of estimates for linear forms in two complex logarithms, which, in partic- ular, imply lower estimates for the difference between integral powers of real algebraic numbers, are given in Chapter 11. Analogous estimates for linear forms in two p-adic logarithms, that is, upper estimates for the p-adic valuation of the difference between in- tegral powers of algebraic numbers are given in Chapter 12. An estimate for linear forms in an arbitrary number of complex logarithms is derived in Chapter 4 from the estimate for linear forms in two complex logarithms established in Chapter 11. While the former estimate is not as strong and general as the estimates stated in Chapter 2, it is sufficiently precise for many applications. We collect open problems in Chapter 13. The thirteen chapters are complemented by six appendices, which, mostly without proofs, gather classical results on approximation by rational numbers, the theory of heights, algebraic number theory, and p-adic analysis. We have tried, admittedly without too much success, to curb our taste for extensive bibliographies. No effort has been made towards exhaustivity, including in the list of bibliographic references, and the topics covered in this textbook reflect somehow the personal taste of the author.
- 11. Preface ix Inevitably, there is some overlap between this monograph and the monograph [376] of Shorey and Tijdeman, which, although over thirty years old, remains an invaluable reference for anyone interested in Diophantine equations. In particular, the content of Chapter 4 (except Section 4.1) is treated in [376] in much greater generality. There is also some overlap with Sprindžuk’s book [386] and the monograph of Evertse and Győry [182]. Regarding the theory of linear forms in logarithms, Chapters 2 and 11 can be seen as an introduction to the book of Waldschmidt [432]. As far as we are aware, the content of Chapters 5, 7, 8, 9, and 12 and several other parts of the present monograph have never appeared in books. To keep this book reasonably short and accessible to graduate and post-graduate students, the results are not proved in their greatest generality and proofs of the best known lower bounds for linear forms in an arbitrary number of complex (resp., p-adic) logarithms are not given. Many colleagues sent me comments, remarks, and suggestions. I am grateful to all of them. Special thanks are due to Samuel Le Fourn, who very carefully read the manuscript and sent me many insightful suggestions. This book was written while I was director of the ‘Institut de Recherche Mathématique Avancée’.
- 13. Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Frequently used notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv 1. Brief introduction to linear forms in logarithms . . . . . . . . . . . . . . . . . 1 1.1. Linear forms in complex logarithms . . . . . . . . . . . . . . . . . . . . 1 1.2. Linear forms in p-adic logarithms . . . . . . . . . . . . . . . . . . . . . 6 1.3. Linear forms in elliptic logarithms . . . . . . . . . . . . . . . . . . . . . 6 2. Lower bounds for linear forms in complex and p-adic logarithms . . . . . . . . 9 2.1. Lower bounds for linear forms in complex logarithms . . . . . . . . . . . 9 2.2. Multiplicative dependence relations between algebraic numbers . . . . . . 17 2.3. Lower bounds for linear forms in p-adic logarithms . . . . . . . . . . . . 17 2.4. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3. First applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1. On the distance between powers of 2 and powers of 3 . . . . . . . . . . . 23 3.2. Effective irrationality measures for quotients of logarithms of integers . . 25 3.3. On the distance between two integral S-units . . . . . . . . . . . . . . . 25 3.4. Effective irrationality measures for n-th roots of algebraic numbers . . . . 26 3.5. On the greatest prime factor of values of integer polynomials . . . . . . . 27 3.6. On the greatest prime factor of terms of linear recurrence sequences . . . 30 3.7. Perfect powers in linear recurrence sequences . . . . . . . . . . . . . . . 32 3.8. Simultaneous Pellian equations and Diophantine quadruples . . . . . . . 35 3.9. On the representation of integers in distinct bases . . . . . . . . . . . . . 39 3.10. Further applications (without proofs) . . . . . . . . . . . . . . . . . . . . 43 3.11. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.12. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4. Classical families of Diophantine equations . . . . . . . . . . . . . . . . . . . 47 4.1. Proof of Baker’s Theorem 1.10 . . . . . . . . . . . . . . . . . . . . . . . 47 4.2. The unit equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.3. The Thue equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.4. Effective improvement of Liouville’s inequality . . . . . . . . . . . . . . 55 4.5. The superelliptic and hyperelliptic equations . . . . . . . . . . . . . . . . 55 4.6. The Diophantine equation x2 C C D yn . . . . . . . . . . . . . . . . . . 57 4.7. Perfect powers at integer values of polynomials . . . . . . . . . . . . . . 60
- 14. xii Contents 4.8. The Catalan equation and the Pillai conjecture . . . . . . . . . . . . . . . 62 4.9. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.10. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5. Further applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.1. Effective irrationality measures for quotients of logarithms of rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.2. Effective irrationality measures for n-th roots of rational numbers . . . . . 68 5.3. The Thue equation axn byn D c . . . . . . . . . . . . . . . . . . . . . 69 5.4. A generalization of Diophantine quadruples . . . . . . . . . . . . . . . . 71 5.5. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.6. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6. Applications of linear forms in p-adic logarithms . . . . . . . . . . . . . . . . 75 6.1. On the p-adic distance between two integral S-units . . . . . . . . . . . . 75 6.2. Waring’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.3. On the b-ary expansion of an algebraic number . . . . . . . . . . . . . . 77 6.4. Repunits and perfect powers . . . . . . . . . . . . . . . . . . . . . . . . 80 6.5. Perfect powers with few binary digits . . . . . . . . . . . . . . . . . . . . 81 6.6. The S-unit equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.7. The Thue–Mahler equation and other classical equations . . . . . . . . . 85 6.8. Perfect powers in binary recurrence sequences . . . . . . . . . . . . . . . 86 6.9. Perfect powers as sum of two integral S-units . . . . . . . . . . . . . . . 88 6.10. On the digital representation of integral S-units . . . . . . . . . . . . . . 89 6.11. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.12. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 7. Primitive divisors and the greatest prime factor of 2n 1 . . . . . . . . . . . . 95 7.1. Primitive divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.2. Primitive divisors of Lucas–Lehmer sequences . . . . . . . . . . . . . . 98 7.3. The Diophantine equation x2 C C D yn , continued . . . . . . . . . . . . 101 7.4. On the number of solutions to the Diophantine equation x2 C D D pn . . 102 7.5. On the greatest prime factor of 2n 1 . . . . . . . . . . . . . . . . . . . 103 7.6. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.7. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 8. The abc-conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 8.1. Effective results towards the abc-conjecture . . . . . . . . . . . . . . . . 109 8.2. Proofs of Theorems 8.2 and 8.3 . . . . . . . . . . . . . . . . . . . . . . . 110 8.3. Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.4. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 9. Simultaneous linear forms in logarithms and applications . . . . . . . . . . . . 117 9.1. A theorem of Loxton . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 9.2. Perfect powers in short intervals . . . . . . . . . . . . . . . . . . . . . . 118 9.3. Simultaneous Pellian equations with at most one solution . . . . . . . . . 121 9.4. Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 9.5. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
- 15. Contents xiii 10. Multiplicative dependence relations between algebraic numbers . . . . . . . . . 125 10.1. Lower bound for the height of an algebraic number . . . . . . . . . . . . 125 10.2. Existence of small multiplicative dependence relations . . . . . . . . . . 127 11. Lower bounds for linear forms in two complex logarithms: proofs . . . . . . . . 131 11.1. Three estimates for linear forms in two complex logarithms . . . . . . . . 131 11.2. An auxiliary inequality involving several parameters . . . . . . . . . . . . 132 11.3. Proof of Theorem 11.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 11.4. Deduction of Theorem 11.1 from Theorem 11.4 . . . . . . . . . . . . . . 138 11.5. Deduction of Theorem 11.2 from Theorem 11.4 . . . . . . . . . . . . . . 140 12. Lower bounds for linear forms in two p-adic logarithms: proofs . . . . . . . . . 143 12.1. Three estimates for linear forms in two p-adic logarithms . . . . . . . . . 143 12.2. Two auxiliary inequalities involving several parameters . . . . . . . . . . 145 12.3. Proofs of Theorems 12.4 and 12.5 . . . . . . . . . . . . . . . . . . . . . 147 12.4. Deduction of Theorem 12.1 from Theorem 12.4 . . . . . . . . . . . . . . 153 12.5. Deduction of Theorem 12.2 from Theorem 12.5 . . . . . . . . . . . . . . 155 12.6. Deduction of Theorem 12.3 from Theorem 12.1 . . . . . . . . . . . . . . 156 13. Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 13.1. Classical conjectures in transcendence theory . . . . . . . . . . . . . . . 157 13.2. Diophantine equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 13.3. Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 A. Approximation by rational numbers . . . . . . . . . . . . . . . . . . . . . . . 167 B. Heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 B.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 B.2. The Liouville inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 176 B.3. Linear forms in one p-adic logarithm . . . . . . . . . . . . . . . . . . . . 177 C. Auxiliary results on algebraic number fields . . . . . . . . . . . . . . . . . . . 179 C.1. Real quadratic fields and Pellian equations . . . . . . . . . . . . . . . . . 179 C.2. Algebraic number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 D. Classical results on prime numbers . . . . . . . . . . . . . . . . . . . . . . . . 183 E. A zero lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 F. Tools from complex and p-adic analysis . . . . . . . . . . . . . . . . . . . . . 191 F.1. The Schwarz lemma in complex analysis . . . . . . . . . . . . . . . . . . 191 F.2. Auxiliary results on p-adic fields . . . . . . . . . . . . . . . . . . . . . . 191 F.3. The Schwarz lemma in p-adic analysis . . . . . . . . . . . . . . . . . . . 193 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
- 17. Frequently used notation log : unless otherwise specified, log z denotes the principal determination of the logarithm of the non-zero complex number z, that is, writing z D rei with r positive and in . ; , we have log z D log rCi. e : base of the natural logarithm. deg : degree (of a polynomial, of an algebraic number). det : determinant. positive : strictly positive. bxc : largest integer x. dxe : smallest integer x. fg : fractional part. k k : distance to the nearest integer. Card : cardinality (of a finite set). perfect power : integer of the form ab , with a 1 and b 2 integers. p1 p2 : the set of all prime numbers ranged in increasing order. q1 qs : a collection of s distinct prime numbers, not necessarily consecutive. P Œn : greatest prime factor of the integer n, with P Œ0 D P Œ˙1 D 1. !.n/ : number of distinct prime factors of the positive integer n, with !.1/ D 0. ' : Euler totient function (Definition D.3). : Möbius function (Definition D.3). ˆd .X; Y / : homogeneous d-th cyclotomic polynomial. h.˛/ : logarithmic Weil height of the algebraic number ˛ (Definition B.4). L.P / : length of the polynomial P.X1; : : : ; Xn/ (Definition B.9). j jp : p-adic absolute value, normalized such that jpjp D p 1 . vp : p-adic valuation, normalized such that vp.p/ D 1. S : finite, non-empty set of prime numbers (or, sometimes, of places of an algebraic number field).
- 18. xvi Frequently used notation ŒnS : S-part of the non-zero integer n, defined by ŒnS D Q p2S jnj 1 p . S-unit : rational number whose numerator and denominator are only com- posed of prime numbers in S. integral S-unit : rational integer being an S-unit. a b : the quantity a is less than b times an absolute, positive, effectively computable real number. a c1;:::;cn b : the quantity a is less than b times an effectively computable, positive, real number, which depends at most on c1; : : : ; cn. a ineff b : the quantity a is less than b times an absolute, positive real number. logp; expp : p-adic logarithm and p-adic exponential functions. SN : set of permutations of f1; : : : ; N g. eK; fK : ramification index and residue degree of a p-adic field K. a; b; c; p : ideals of an algebraic number field. ep; fp : ramification index and residue degree of an ideal p. : Galois conjugacy. K, OK, O K, MK, M1 K , DK, hK : an algebraic number field, its ring of integers, group of units, set of places, set of infinite places, discriminant, and class number.
- 19. Chapter 1 Brief introduction to linear forms in logarithms We start this textbook with a concise introduction to the theory of linear forms in logarithms. For much more comprehensive historical surveys and many bibliographic references, the reader is directed to the introductory text [441] and the monographs [28, 38, 141, 243, 419, 432]. Throughout, unless otherwise specified, log z denotes the principal determination of the logarithm of the non-zero complex number z, that is, writ- ing z D rei with r positive and in . ; , we have log z D log r C i. In particular, log. 1/ is equal to i. 1.1. Linear forms in complex logarithms A complex number is called algebraic if it is a root of a non-zero polynomial with integer coefficients. A complex number which is not algebraic is called transcendental. Rational numbers and their integer roots are obvious examples of algebraic numbers. The first examples of transcendental numbers were given, and even in a totally explicit form, by Liouville [264] in 1844, in a note where he proved that a real algebraic irrational number cannot be too close, in a suitable sense, to rational numbers; see Theorem A.5 for a precise formulation. It is one thing to construct transcendental numbers; it is another, much more difficult one, to prove the transcendence of some explicitly given complex numbers. The first result in this direction is the proof of the transcendence of e, obtained by Hermite in 1873. Shortly thereafter, in 1882, Lindemann established that is transcendental. The now famous Hermite–Lindemann theorem reads as follows. Theorem 1.1. For any non-zero complex number ˇ, at least one of the two numbers ˇ and eˇ is transcendental. Theorem 1.1 includes the transcendence of e (take ˇ D 1) and that of (take ˇ D i). It also shows that log 2 is transcendental and, more generally, that any determination of the logarithm of an algebraic number different from 0 and 1 is transcendental. In 1885 Weierstrass extended Theorem 1.1 as follows. Theorem 1.2. Let n 2 be an integer. Let ˛1; : : : ; ˛n be distinct algebraic numbers. Then, e˛1 ; : : : ; e˛n are linearly independent over the field of algebraic numbers. Sincemonomialsine˛1 ; : : : ;e˛n areexponentialsofintegralcombinationsof ˛1; : : : ; ˛n, it is easy to show that Theorem 1.2 is equivalent to the following statement: If n 2
- 20. 2 1. Brief introduction to linear forms in logarithms and ˛1; : : : ; ˛n are algebraic numbers that are linearly independent over the field of ratio- nal numbers, then e˛1 ; : : : ; e˛n are algebraically independent over the field of algebraic numbers (this means that no non-zero polynomials with algebraic coefficients vanish at .e˛1 ; : : : ; e˛n /). In 1900, David Hilbert proposed a list of twenty-three open problems and presented ten of them in Paris at the second conference of the International Congress of Mathematicians. His seventh problem is the following (observe that e D . 1/ i ): The expression ˛ˇ for an algebraic base ˛ different from 0 and 1 and an irrational algebraic exponent ˇ, e.g. the number 2 p 2 or e , always represents a transcendental or at least an irrational number. This problem was solved in 1934 independently and simultaneously by Gelfond [197] and Schneider [359], by different methods. Theorem 1.3. For any non-zero algebraic numbers ˛1; ˛2; ˇ1; ˇ2 with log ˛1 and log ˛2 linearly independent over the rationals, we have ˇ1 log ˛1 C ˇ2 log ˛2 6D 0: Theorem 1.3 was generalized to linear combinations of n logarithms of algebraic numbers by Baker [16,17] in 1966 and 1967. Theorem 1.4. Let n 2 be an integer. Let ˛1; : : : ; ˛n be non-zero algebraic numbers and log any fixed determination of the logarithm function. If log ˛1; : : : ; log ˛n are linearly independent over the rationals, then they are linearly independent over the field of algebraic numbers. Shortly thereafter, Baker [18] extended his previous result as follows. Theorem 1.5. Let n be a positive integer. Let ˛1; : : : ; ˛n be non-zero algebraic numbers and log ˛1; : : : ; log ˛n any determinations of their logarithms. If log ˛1; : : : ; log ˛n are linearly independent over the rationals, then 1; log ˛1; : : : ; log ˛n are linearly independent over the field of algebraic numbers. It readily follows from Theorem 1.5 that the complex number eˇ0 ˛ˇ1 1 ˛ˇn n is tran- scendental for all non-zero algebraic numbers ˛1; : : : ; ˛n, ˇ0; : : : ; ˇn. Furthermore, Theorem 1.5 includes Theorem 1.1. Theorems 1.3 to 1.5 show that any expression of the form ˇ0 C ˇ1 log ˛1 C C ˇn log ˛n; (1.1) where ˛1; : : : ; ˛n, ˇ1; : : : ; ˇn are non-zero algebraic numbers and ˇ0 is algebraic, vanishes only in trivial cases. A natural question is then to bound from below its absolute value (when non-zero). For the sake of simplification, we assume in the discussion below that the algebraic numbers involved are all rational numbers. Let n 2 be an integer. For j D 1; : : : ; n, let xj yj be a non-zero rational number, bj a non-zero integer, and set B WD maxf3; jb1j; : : : ; jbnjg and Aj WD maxf3; jxj j; jyj jg: (1.2)
- 21. 1.1. Linear forms in complex logarithms 3 We consider the rational number ƒ WD x1 y1 b1 xn yn bn 1: (1.3) Since we wish to bound jƒj from below, we may assume that jƒj 1 2 . Then, the linear form in logarithms of rational numbers , defined by WD log.1 C ƒ/ D b1 log x1 y1 C C bn log xn yn ; satisfies jƒj 2 jj 2jƒj: A trivial estimate of the denominator of (1.3) gives that ƒ D 0 or log jƒj n X jD1 jbj j log maxfjxj j; jyj jg B n X jD1 log Aj : (1.4) The dependence on the Aj ’s in (1.4) is very satisfactory, unlike the dependence on B. For applications to Diophantine problems, we require a better estimate in terms of B than the one given in (1.4), even if it comes with a weaker one in terms of the Aj ’s. For example, replacing B by o.B/ is sufficient in many cases (see, for example, Theorems 3.10, 3.13, and 4.9), but not in all cases (see, for example, Theorems 3.3, 3.4, and 5.1). The next lemma shows that B cannot be replaced by o.log B/. Lemma 1.6. Let n; a1; : : : ; an be integers, all of which are greater than or equal to 2. Set A D maxfa1; : : : ; ang. Then, for every integer B greater than 2n log A, there exist rational integers b1; : : : ; bn with 0 maxfjb1j; : : : ; jbnjg B and jab1 1 abn n 1j 2n log A Bn 1 : Lemma 1.6 is a direct consequence of the Dirichlet Schubfachprinzip applied to the points b1 log a1 C C bn log an with 0 b1; : : : ; bn B, which all lie in the interval Œ0; nB log A. The first effective improvement of (1.4) was obtained by Gelfond [198] in 1935 in the case n D 2. He proved that, for multiplicatively independent positive rational numbers x1 y1 ; x2 y2 , for an arbitrary positive real number , and for all integers b1; b2, not both 0, we have ˇ ˇ ˇ x1 y1 b1 x2 y2 b2 1 ˇ ˇ ˇ x1 y1 ; x2 y2 ; exp .log B/5C ; where B D maxf3; jb1j; jb2jg. Throughout this book, the notation a c1;:::;cn b (resp., a ineff c1;:::;cn b) means that the quantity a is greater than b times an effectively computable positive real number (resp., a positive real number), which depends at most
- 22. 4 1. Brief introduction to linear forms in logarithms on c1; : : : ; cn. Subsequently, Gelfond improved his own result and totally explicit estimates were provided by Schinzel [354] in 1967. Gelfond [200] also gave an estimate valid for a linear form in an arbitrary number of logarithms. Theorem 1.7. Let n 2 be an integer and a1; : : : ; an positive rational numbers which are multiplicatively independent. Let ı be a positive real number. Let b1; : : : ; bn be rational integers, not all zero, and set B D maxf3; jb1j; : : : ; jbnjg. Then, we have jab1 1 abn n 1j ineff n;a1;:::;an; ı exp. ıB/: The proof of Theorem 1.7, which rests on a theorem of Siegel, does not enable us to compute effectively the implicit numerical constant. In his book Gelfond [200] pointed out the importance of getting an effective version of Theorem 1.7. For n 3, the first non-trivial effective lower bound for the quantity ƒ defined in (1.3) was given by Baker [16] in 1966. With B as in (1.2), he obtained that either ƒ D 0 or, for every positive real number , we have ˇ ˇ ˇ x1 y1 b1 xn yn bn 1 ˇ ˇ ˇ n; x1 y1 ;:::; xn yn ; exp .log B/nC1C : His result is much more general and applies to expressions of the form (1.1). Shortly thereafter, Feldman [185,186] established the following refinement of Baker’s lower bound. Theorem 1.8. Let n 2 be an integer and a1; : : : ; an positive rational numbers which are multiplicatively independent. Let b1; : : : ; bn be rational integers, not all zero, and set B D maxf3; jb1j; : : : ; jbnjg. Then, there exists a positive, effectively computable real number C, depending only on n; a1; : : : ; an, such that jab1 1 abn n 1j exp. C log B/ D B C : (1.5) Lemma 1.6 shows that the dependence on B in the estimate (1.5) is essentially best possible, but, for applications, it is much desirable to determine precisely how C depends on the rational numbers a1; : : : ; an. A first explicit result in this direction was given by Baker [19] in 1968. Throughout this chapter, the height of an algebraic number is the naïve height, that is, the maximum of the absolute values of the coefficients of its minimal defining polynomial over the integers. We reproduce below a consequence of the main theorem of [19]. Theorem 1.9. Assume that n 2 and that ˛1; : : : ; ˛n are non-zero algebraic numbers, whose degrees do not exceed D and whose heights do not exceed A, where D 4 and A 4. Let ı be a real number with 0 ı 1. Let b1; : : : ; bn be rational integers, not all zero, and set B D maxf3; jb1j; : : : ; jbnjg. If j˛b1 1 ˛bn n 1j e ınB 2 ; then B .4.nC1/2 ı 1 D2.nC1/ log A/.2nC3/2 :
- 23. 1.1. Linear forms in complex logarithms 5 In many applications, especially to classical families of Diophantine equations, we are led to bound from below expressions of the form j˛b1 1 ˛bn n ˛nC1 1j, where ˛nC1 has a large height; see Chapter 4. In this respect, Theorem 1.9 is not plainly satisfactory, since ˛1; : : : ; ˛n play the same rôle, irrespective of their height and their exponent. Its following refinement, established by Baker [26], appears to be very useful. Theorem 1.10. Assume that n 1 and let ˛1; : : : ; ˛nC1 be non-zero algebraic numbers of degree at most D. Let the heights of ˛1; : : : ; ˛n and ˛nC1 be at most A and AnC1 respectively, where A 2 and AnC1 2. Let ı be a positive real number. Let b1; : : : ; bn be integers, not all zero, and set B D maxf3; jb1j; : : : ; jbnjg. If 0 j˛b1 1 ˛bn n ˛nC1 1j e ıB ; then B n;D;A;ı log AnC1: In the present textbook, we give a complete proof of Theorem 1.10. We derive it from a lower bound for linear forms in two logarithms; see Section 4.1. In 1975, Baker [27] established a subsequent refinement of Theorem 1.8. Theorem 1.11. Assume that n 2 and that ˛1; : : : ; ˛n are non-zero algebraic numbers, whose degrees do not exceed D. Assume that, for j D 1; : : : ; n, the height of ˛j does not exceed Aj , where Aj 2. Let b1; : : : ; bn be rational integers, not all zero, and set B D maxf3; jb1j; : : : ; jbnjg. If ˛b1 1 ˛bn n is not equal to 1, then j˛b1 1 ˛bn n 1j B C… log … ; (1.6) where … D log A1 log An and C is an effectively computable real number depending only on n and D. As was pointed out by Baker below the statement of his theorem, it would be of much interest to eliminate log … in (1.6). Also, Theorem 1.11 does not include Theorems 1.9 and 1.10, nor the main result of Baker’s paper [25]. The deletion of the log … factor (which is mainly interesting from a theoretical point of view, but also has several applications, see, for example, Theorem 3.6) has been achieved independently by Wüstholz [440] and by Philippon and Waldschmidt [329]. Thus, at the end of the ’80s, it was established that, with ƒ, A1; : : : ; An, and B as in (1.2) and (1.3), there exists an effectively computable real number c.n/, depending only on the number n of rational numbers involved in (1.3), such that the lower estimate log ˇ ˇ ˇ x1 y1 b1 xn yn bn 1 ˇ ˇ ˇ c.n/ log A1 log An log B (1.7) holds, when ƒ is non-zero. Since then, several authors have managed to considerably reduce the value of the real number c.n/. However, it remains an open problem to replace the product log A1 log An by the sum log A1 C C log An; see Chapter 13.
- 24. 6 1. Brief introduction to linear forms in logarithms 1.2. Linear forms in p-adic logarithms Let p be a prime number. In parallel with the development of the theory of linear forms in complex logarithms, progress has been regularly made towards its p-adic analogue. For a non-zero rational number x, let vp.x/ denote the exponent of p in the decompo- sition of x as a product of prime powers. With ƒ; A1; : : : ; An, and B as in (1.2) and (1.3), a trivial estimate shows that vp x1 y1 b1 xn yn bn 1 log.2Ab1 1 Abn n / log p 1 C B log p n X j D1 log Aj : (1.8) The theory of p-adic linear forms in logarithms gives a much better result in terms of the dependence on B, namely, that there exists an effectively computable real number c0 .n/, depending only on the number n of rational numbers involved, such that the upper estimate vp x1 y1 b1 xn yn bn 1 c0 .n/ p log A1 log An log B (1.9) holds, when ƒ is non-zero. In terms of B, this is much better than (1.8) and analogous to (1.7). One of the major open problems in the theory is to remove (or at least to improve) the dependence on p in (1.9), which remains very unsatisfactory and is ultimately a consequence of the fact that the radius of convergence of the p-adic exponential function is finite. A good reference for an historical introduction is [448]. Mahler [278] established in 1932 the p-adic analogue of the Hermite–Lindemann theorem and three years later [280] the p-adic analogue of the Gelfond–Schneider theorem. Gelfond [199] proved a quan- titative estimate for linear forms in two p-adic logarithms, which was later refined by Schinzel [354]. Estimates for linear forms in an arbitrary number of p-adic logarithms were obtained by Brumer [99], Sprindžuk [384,385], Coates [149], Kaufman [238], Baker and Coates [34], van der Poorten [335], Dong [168,169], and Yu [443,445–447], among others. 1.3. Linear forms in elliptic logarithms A few years after the birth of the theory of linear forms in logarithms, it was realized that the techniques used in the proofs can be applied to any commutative algebraic group. Initial steps towards the derivation of elliptic analogues were made by Baker [23], who considerably extended earlier results of Schneider [360]. Elliptic curves with complex mul- tiplication were studied by Masser [284] in 1975 and, a year later, Coates and Lang [151] established the first lower bounds in the case of Abelian varieties with complex multi- plication. Subsequently, Philippon and Waldschmidt [330] and Hirata-Kohno [229,230] obtained rather general statements. In the case of elliptic logarithms, the first totally explicit estimates were given by David [162]. These have applications to the complete determination of the integral points on an elliptic curve, as is very well explained by Stroeker and Tzanakis [408], Gebel, Pethő, and Zimmer [196], and Tzanakis [420,421].
- 25. 1.3. Linear forms in elliptic logarithms 7 Let n be a positive integer and E1; : : : ; En elliptic curves over an algebraic number field K. For j D 1; : : : ; n, consider a Weierstrass model of Ej and }j the associated Weierstrass function. Let uj be a complex number such that }j .uj / is in K [ f1g. Such a complex number uj is an elliptic logarithm of an algebraic point of Ej . Also, let ˇ0; ˇ1; : : : ; ˇn be elements of K. Let B 3 denote an upper bound for the heights of ˇ0; ˇ1; : : : ; ˇn. Set ƒe WD ˇ0 C ˇ1u1 C C ˇnun: David and Hirata-Kohno [163] established that there exists an effectively computable positive real number c, which depends only on K; n; u1; : : : ; un and the curves E1; : : : ; En, such that we have jƒej B c if ƒe is non-zero. For the state-of-the-art and generalisations to Abelian varieties, the reader is directed to [163] and to Gaudron’s papers [192–194]. Lower bounds for linear forms in p-adic elliptic logarithms were given by Rémond and Urfels [345] and Hirata-Kohno and Takada [232]; see also Fuchs and Pham [189].
- 27. Chapter 2 Lower bounds for linear forms in complex and p-adic logarithms We list below several estimates for linear forms in complex and p-adic logarithms, which will be used throughout the book. We give only few bibliographic references and direct the reader to the monograph of Waldschmidt [432] for further information, in particular to its Section 10.4. Most of the results quoted are corollaries of more precise estimates, so the reader wishing to apply the theory of linear forms in logarithms should better consult the original papers to find the sharpest bounds available to date. In this chapter we write completely explicit estimates. This will, however, not be the case for most of the results presented in the next chapters, where we will often make no effort to give explicit values for the numerical constants. Throughout, h denotes the (logarithmic) Weil height; see Definition B.4. 2.1. Lower bounds for linear forms in complex logarithms We start with a general theorem of Waldschmidt [430,432] on inhomogeneous linear forms in logarithms of algebraic numbers with algebraic coefficients. Theorem 2.1. Let n 1 be an integer. Let ˛1; : : : ; ˛n be non-zero algebraic numbers. Let log ˛1; : : : ; log ˛n be determinations of their logarithm and assume that log ˛1; : : : ; log ˛n are linearly independent over the rationals. Let ˇ0; : : : ; ˇn be algebraic numbers, not all zero. Let D be the degree over Q of the number field Q.˛1; : : : ; ˛n; ˇ0; : : : ; ˇn/. Let E; E , and A1; : : : ; An be real numbers with E E1=D e1=D ; E e; E D log E ; and log Aj max n h.˛j /; E D j log ˛j j; log E D o ; 1 j n: Let B be a real number with B E ; B max 1jn D log Aj log E ; log B max 0j n h.ˇj /:
- 28. 10 2. Lower bounds for linear forms in complex and p-adic logarithms Then, we have log jˇ0 C ˇ1 log ˛1 C C ˇn log ˛nj 2nC25 n3nC9 DnC2 log A1 : : : log An log B log E .log E/ n 1 : Assume that we are in the homogeneous rational case, that is, assume that ˇ0 D 0 and ˇ1; : : : ; ˇn are rational integers b1; : : : ; bn with bn ¤ 0. Let B0 be a real number satisfying B0 E ; B0 max 1j n 1 n jbnj log Aj C jbj j log An o : Then, we have log jb1 log ˛1 C C bn log ˛nj 2nC26 n3nC9 DnC2 log A1 : : : log An log B0 log E .log E/ n 1 : In particular, choosing E D e and E D 3D, we get log jb1 log ˛1 C C bn log ˛nj 2nC26 n3nC9 DnC2 log.3D/ log A1 : : : log An log B0 : Proof. This follows from Theorem 9.1 of [432], taking into account Remark 3 on page 303 and Proposition 9.18 of [432]. By Proposition 9.21 of [432], the assumption in Theorem 2.1 that log ˛1; : : : ; log ˛n are linearly independent over the rationals can be relaxed to the assumptions ˇ0 Cˇ1 log ˛1 C C ˇn log ˛n ¤ 0 and D3 .log B0/.log Aj /.log E / .log D/.log E/2 ; 1 j n; where B0 D B in the general case and B0 D B0 in the homogeneous rational case. Theorem 2.1 plainly includes Theorem 1.5. It also contains Theorem 1.11 and the lower bound (1.7). Taking n 2 and bn D 1 in the homogeneous rational case of Theorem 2.1 and setting B WD maxf3; jb1j; : : : ; jbn 1jg, we get log jb1 log ˛1 C C bn 1 log ˛n 1 C log ˛nj n;D log A1 : : : log An log B log An : This has many important applications, in particular to Diophantine equations; see Theo- rems 4.1, 4.3, and 4.5. Assuming that there exists a real number ı such that 0 ı 1 2 and jb1 log ˛1 C C bn 1 log ˛n 1 C log ˛nj e ıB , we deduce that B n;D ı 1 log A1 : : : log An 1 log ı 1 log.Dn 1 A1 : : : An 1/ log An: This (and the discussion below explaining how the linear form (2.1) and the quantity (2.2) are related) shows that Theorem 2.1 also contains Theorem 1.10. In the homogeneous rational case, a result similar to Theorem 2.1 was proved in- dependently by Baker and Wüstholz [37, 38]. Since their estimate (which has a better
- 29. 2.1. Lower bounds for linear forms in complex logarithms 11 dependence on n than in Theorem 2.1, namely the factor n3n is replaced by n2n ) does not include the useful parameters E and B0 , and is superseded by Theorem 2.2 below, we do not quote it. The parameter E originates in papers by Shorey [367, 368] and is of interest when ˛1; : : : ; ˛n are real and very close to 1, in which case it can be chosen to be very large. By assumption, log E cannot exceed D min1j n log Aj . However, in some cases, it can be taken close to this quantity. To see this in the homogeneous rational case, assume that, for j D 1; : : : ; n, we have ˛j D 1 C 1 xj , for an integer xj 3. Then, setting E D E D min 1j n xj and Aj D xj C 1; 1 j n; we see that, since B0 E, Theorem 2.1 implies the lower bound log jb1 log ˛1 C Cbn log ˛nj n log A1 : : : log An .min1j n log Aj /n 1 log maxf3; jb1j; : : : ; jbnjg: In the most favourable cases, for example when there exists a real number M such that max1j n xj .min1j n xj /M , we get log jb1 log ˛1 C C bn log ˛nj n Mn 2 max 1j n log Aj log maxf3; jb1j; : : : ; jbnjg; thus replacing the product of the log Aj , as it occurs in the statement of Theorem 2.1, with their maximum. Further explanations are given below Theorem 2.5, in Chapter 5, and in Section 10.4.3 of [432]. In the course of this textbook, we mention in passing a few applications of the first assertion of Theorem 2.1 (at the beginning of Section 3.3 and in Section 3.10), but we only apply estimates for homogeneous linear forms in logarithms with integer coefficients. Therefore, we focus our attention on the second assertion of Theorem 2.1 and its subsequent improvements and refinements. Let n 2 be an integer. Let ˛1; : : : ; ˛n be non-zero algebraic numbers. Let b1; : : : ; bn be integers. Let log ˛1; : : : ; log ˛n be any determination of the logarithms of ˛1; : : : ; ˛n. The theory of linear forms in logarithms provides us with lower bounds for the absolute value of the linear form b1 log ˛1 C C bn log ˛n; (2.1) when it is non-zero. This yields lower bounds for the quantity j˛b1 1 : : : ˛bn n 1j; (2.2) which occurs frequently in Diophantine questions. We choose below to mainly consider quantities of the form (2.2) and not (2.1). When ˛1; : : : ; ˛n are all real numbers, both forms are essentially equivalent since log.1 C x/ D x C O.x2 / in the neighborhood of the origin. This is however not the case when complex non-real numbers are among ˛1; : : : ; ˛n. Indeed, denoting by log the principal determination of the logarithm, to say that (2.2) is small does not imply that b1 log ˛1 C C bn log ˛n is close to 0, but merely
- 30. 12 2. Lower bounds for linear forms in complex and p-adic logarithms that b1 log ˛1 C C bn log ˛n is close to an integer multiple of 2i. Precisely, we derive the existence of an integer b0 with jb0j jb1j C C jbnj and such that the linear form b0 log. 1/ C b1 log ˛1 C C bn log ˛n is close to 0. The next statement is a corollary of the, at present time, best known general estimate, due to Matveev [290, 291]. The crucial improvement on the earlier results [37, 430] concerns the dependence on the number n of logarithms: it is exponential in n, and not of the form ncn as in the earlier estimates. Theorem 2.2. Let n 1 be an integer. Let ˛1; : : : ; ˛n be non-zero algebraic numbers. Let D be the degree over Q of a number field containing ˛1; : : : ; ˛n. Let A1; : : : ; An be real numbers with log Aj max n h.˛j /; j log ˛j j D ; 0:16 D o ; 1 j n: Let b1; : : : ; bn be integers and set B D maxfjb1j; : : : ; jbnjg and B00 D max n 1; max n jbj j log Aj log An W 1 j n oo : Then, we have log j˛b1 1 : : : ˛bn n 1j 330nC4 .nC1/5:5 DnC2 log.eD/ log A1 : : : log An log.enB/ (2.3) and, if n 2 and ˛1; : : : ; ˛n are all real numbers, we get the better lower bound log j˛b1 1 : : : ˛bn n 1j 2 30nC3 n4:5 DnC2 log.eD/ log A1 : : : log An log.eB/: (2.4) The same statements hold with B replaced by maxfB00 ; nB=.D log An/g in (2.3) and with B replaced by B00 in (2.4). Proof that Theorem 2.2 follows from Matveev’s results. Inequality (2.4) with B replaced or not by B00 is an immediate consequence of Corollary 2.3 of Matveev [291]. Denote by log the principal determination of the logarithm. If j˛b1 1 : : : ˛bn n 1j 1 2 , then there exists an integer b0, with jb0j n B, such that WD jb0 log. 1/ C b1 log ˛1 C C bn log ˛nj satisfies j˛b1 1 : : : ˛bn n 1j 2 . Noticing that j log. 1/j D and h. 1/ D 0, we set log A0 D D and deduce (2.3) from Corollary 2.3 of Matveev [291]. Since jb0j log A0 log An nB D log An ; we see that B can be replaced by maxfB00 ; nB=.D log An/g in (2.3).
- 31. 2.1. Lower bounds for linear forms in complex logarithms 13 We stress that the definition of B0 in Theorem 2.1 is slightly different from that of B00 in Theorem 2.2. The lower bound (2.4) with B00 in place of B is crucial for the proof of Theorem 3.8. Many Diophantine problems can be reduced to lower bounds for linear forms in two or three logarithms. While, in the case of three logarithms, we do not have very satisfactory estimates (but see [298]), Laurent, Mignotte, and Nesterenko [254] obtained in 1995 rather sharp lower bounds for linear forms in two logarithms. The quality of their result is an illustration of the method of interpolation determinants, which was introduced in this context by Laurent [252]. The current best known estimates have been established by Laurent in 2008 in [253]. We display below an estimate obtained in Corollary 1 of [253] and three consequences of the main result of [254]. Theorem 2.3. Let ˛1 and ˛2 be multiplicatively independent algebraic numbers. Set D0 D ŒQ.˛1; ˛2/ W Q=ŒR.˛1; ˛2/ W R. Let A1 and A2 be real numbers such that log Aj max n h.˛j /; 1 D0 ; j log ˛j j D0 o ; j D 1; 2: Let b1 and b2 be integers, not both zero, and set log B0 D max n log jb1j D0 log A2 C jb2j D0 log A1 C 0:21; 20 D0 ; 1 o : Then, we have the lower bound log jb1 log ˛1 C b2 log ˛2j 25:2 D04 .log A1/.log A2/.log B0 /2 : (2.5) For n D 2, the numerical constant in (2.4) exceeds 109 . It has been substantially reduced in Theorem 2.3. This is crucial for applications to the complete resolution of Diophantine equations. Very roughly speaking, when a Diophantine problem can be reduced to linear forms in only two logarithms, then it can (often, this is not always true! See Problem 13.12) be completely solved. A weaker version (weaker only in terms of the numerical constants) of Theorem 2.3 is proved in this book; see Theorem 11.1. Note also that, unlike in Theorem 2.3, the algebraic numbers ˛1 and ˛2 are not assumed to be multiplicatively independent in Theorem 11.1. This allows us to deduce directly from Theorem 2.3 a slightly weaker version of Theorem 2.6, see Theorem 11.3. Observe that the dependence on B0 in Theorem 2.3 is not best possible and worse than in Theorems 2.1 and 2.2, since log B0 occurs squared. Gouillon [202] established a lower bound for jb1 log ˛1 C b2 log ˛2j, where the dependence on B0 occurs through a factor .log B0 / only. His numerical constants being not so small, it is better for most of the applications to use the bounds established in [253, 254]. However, to have the best possible dependence in B0 is essential for the proofs of Theorems 3.3 and 5.1. As in Theorem 2.1, an extra parameter E was introduced in [254]. We reproduce below Corollaire 3 of [254]. Theorem 2.4. Let ˛1 and ˛2 be multiplicatively independent positive real algebraic numbers. Let D be the degree of the number field Q.˛1; ˛2/. Let A1 and A2 be real
- 32. 14 2. Lower bounds for linear forms in complex and p-adic logarithms numbers such that log Aj max n h.˛j /; 1 D ; j log ˛j j D o ; j D 1; 2: Let b1 and b2 be integers, not both zero. Let E be a real number with E 1 C min nD log A1 j log ˛1j ; D log A2 j log ˛2j o (2.6) and set log B0 D max n log jb1j D log A2 C jb2j D log A1 C log log E C 0:47; 10 log E D ; 1 2 o : Assume furthermore that 2 E minfA3D=2 1 ; A3D=2 2 g. Then, log jb1 log ˛1 C b2 log ˛2j 35:1D4 .log A1/.log A2/.log B0 /2 .log E/ 3 : In [254], the authors defined the parameter E to be equal to the right hand side of (2.6). However, it easily follows from the proof that Theorem 2.4 as stated is correct. A weaker version (weaker only in terms of the numerical constants) of Theorem 2.4 is proved in this book; see Theorem 11.2. The following consequence of Theorem 2.4 emphasizes the rôle of the parameter E in a particular case which occurs frequently in applications. Theorem 2.5. Let x1; x2; y1; y2 be positive integers with x1 2, x1 ¤ y1, and y2 x2 6 5 y2. Let b be a positive integer and assume that .x1 y1 /b ¤ x2 y2 . Define the parameter by x2 y2 D 1 C x 2 : Then, we have log x2 1 and log ˇ ˇ ˇ y1 x1 b x2 y2 1 ˇ ˇ ˇ 35:2 .log x1/ max n 1 C log b log x2 ; 10 o2 : (2.7) Since x2 y2 1, we always have 1. The estimate is stronger when is very close to 1, that is, when x2 y2 is very close to 1. Under the assumption of Theorem 2.5, we get instead of (2.5) a lower bound of the form .log A1/.log A2/ log.x2 y2 1/ .log B0 /2 ; with A1 D maxfx1; 3g and A2 D maxfx2; 3g. This crucial improvement upon the “classical” estimate (2.5) turns out to have many spectacular applications, some of which being given in Chapter 5; see Theorems 5.2, 5.4, and 5.6. Proof of Theorem 2.5 assuming Theorem 2.4. If y1 x1, then the left hand side of (2.7) exceeds log.y1 x1 1/, thus it exceeds log x1 and (2.7) holds. If y1 x1 and x1 5, then .y1 x1 /b 4 5 and (2.7) holds since x2 y2 6 5 .
- 33. 2.1. Lower bounds for linear forms in complex logarithms 15 If x1 y1 and x1 y1 and x2 y2 are multiplicatively dependent, then there exist coprime positive integers c; d and positive integers u; v such that x1 y1 D . c d /u and x2 y2 D . c d /v . We then get ˇ ˇ ˇ y1 x1 b x2 y2 1 ˇ ˇ ˇ D ˇ ˇ ˇ d c ub v 1 ˇ ˇ ˇ 1 maxfc; dg 1 x1 ; and the theorem holds. Thus, we assume that x1 y1 and x2 y2 are multiplicatively independent, x1 y1, and x1 6. Observe that (2.7) holds if x1 x 2 =2. To see this, it suffices to observe that, since y1 x1, we then get 0 y1 x1 b x2 y2 1 1 x1 1 C 1 x 2 1 1 2x1 ; which implies that ˇ ˇ ˇ y1 x1 b x2 y2 1 ˇ ˇ ˇ 1 2x1 : Thus, in the sequel, we assume that x 2 2x1. We apply Theorem 2.4 with ˛1 D x1 y1 and ˛2 D x2 y2 . Then, D D1, A1 Dx1 and A2 Dx2 (note that the assumption y2 x2 6 5 y2 implies that x2 6). Furthermore, we set E D x 2 : Observe that E 5, since 5x2 6y2. This proves the first assertion of the theorem. Furthermore, we have E minfx3=2 1 ; x3=2 2 g, since x1 6 and 1. We deduce from log.1 C x 2 / x 2 that 1 C log x2 log x2 y2 D 1 C log x2 log.1 C x 2 / 1 C x 2 log x2 E: Furthermore, we may assume that ˇ ˇ ˇ y1 x1 b x2 y2 1 ˇ ˇ ˇ x 10 1 ; since otherwise the theorem clearly holds. Using this inequality and 5 x 2 2x1, we deduce that b log x1 y1 log x2 y2 C 2x 10 1 x 2 C 211 x 10 2 2x 2 ; thus 1 C log x1 log x1 y1 1 C b 2 x 2 log x1 E; since x1 6. Consequently, we have checked that E 1 C min nlog x1 log x1 y1 ; log x2 log x2 y2 o :
- 34. 16 2. Lower bounds for linear forms in complex and p-adic logarithms The assumptions of Theorem 2.4 are then satisfied and we derive that log ˇ ˇ ˇlog x2 y2 b log x1 y1 ˇ ˇ ˇ 35:1 .log x1/ log x2 log E log B0 log E 2 35:1 .log x1/ log B0 log E 2 ; where log B0 log E D max ( log b log x2 C 1 log x1 log E C log log E C 0:47 log E ; 10 ) max n 1 C log b log x2 ; 10 o : We conclude by using that every real number z with jzj 1 2 satisfies j log.1 C z/j 2jzj. This completes the proof of the theorem. For some applications, we need a lower bound for quantities of the shape j˛b 1j, where ˛ is a complex algebraic number of modulus 1. Such an estimate does not follow from Theorem 2.3, since in its statement ˛1 and ˛2 are assumed to be multiplicatively independent. We display a consequence of Théorème 3 of [254]. Theorem 2.6. Let ˛ be a complex algebraic number of modulus 1 which is not a root of unity. Let b be a positive integer. Set D0 D ŒQ.˛/ W Q 2 ; log A D max n 20 D0 ; 11 j log ˛j D0 C h.˛/ o ; and log B0 D max n 17 D0 ; 1 10 p D0 ; log b 25 C 2:35 C 5:1 D0 o : Then, we have log j˛b 1j 9.D0 /3 .log A/.log B0 /2 : Proof that Theorem 2.6 follows from Théorème 3 of [254]. Recall that log denotes the principal determination of the logarithm. If j˛b 1j 1 3 , then there exists an integer b0 with jb0j b such that j˛b 1j jb0 log. 1/ C b log ˛j 2 : We then use the inequalities 1 2D0 log A C 1 68:9 1 40 C 1 68:9 1 25 to get from Théorème 3 of [254] that log jb0 log. 1/ C b log ˛j 8:87.D0 /3 .log A/.log B0 /2 : This yields the desired estimate.
- 35. 2.2. Multiplicative dependence relations between algebraic numbers 17 2.2. Multiplicative dependence relations between algebraic numbers The results displayed in the previous section provide us with lower bounds for the quantity jb1 log ˛1 C C bn log ˛nj; when it is non-zero. However, in some situations, we derive linear forms in logarithms that are equal to zero. It is then often useful to find rational integers b0 1; : : : ; b0 n, not all zero, with small absolute values and such that b0 1 log ˛1 C C b0 n log ˛n D 0: A result of this type was given by Loxton and van der Poorten [271]. We quote below a version of Waldschmidt [432]. Theorem 2.7. Let m 2 be an integer and ˛1; : : : ; ˛m multiplicatively dependent non- zero algebraic numbers. Let log ˛1; : : : ; log ˛m be any determination of their logarithms. Let D bethedegreeofthe numberfieldgeneratedby ˛1; : : : ; ˛m over Q. Forj D 1; : : : ; m, let Aj be a real number satisfying log Aj max n h.˛j /; j log ˛j j D ; 1 o : Then there exist rational integers n1; : : : ; nm, not all zero, such that n1 log ˛1 C C nm log ˛m D 0 and jnj j 11.m 1/D3 m 1 .log A1/ .log Am/ log Aj ; for j D 1; : : : ; m. A full proof of Theorem 2.7, together with additional references, is given in Chapter 10. Theorem 2.7 is used in the proof of Theorem 3.16. 2.3. Lower bounds for linear forms in p-adic logarithms Let p be a prime number and K an algebraic number field. Let OK denote the ring of integers of K. Let p be a prime ideal in K, lying above the prime number p, and denote by ep its ramification index, that is, the exponent of p in the decomposition of the ideal pOK in a product of prime ideals; see Section B.1. For a non-zero algebraic number ˛ in K, let vp.˛/ denote the exponent of p in the decomposition of the fractional ideal ˛OK in a product of prime ideals and set vp.˛/ D vp.˛/ ep : This defines a valuation vp on K which extends the p-adic valuation vp on Q normalized in such a way that vp.p/ D 1.
- 36. 18 2. Lower bounds for linear forms in complex and p-adic logarithms Let ˛1; : : : ; ˛n be elements of K and b1; : : : ; bn non-zero rational integers. We look for an upper bound for the quantity vp.˛b1 1 : : : ˛bn n 1/; where b1; : : : ; bn denote rational integers, not all zero, when ˛b1 1 : : : ˛bn n is not equal to 1. We begin with an easy estimate. Theorem 2.8. Let p be a prime number, b 2 an integer, and ˛ an algebraic number of degree D. If ˛b is not equal to 1, then vp.˛b 1/ log b log p C 2D pD 1 log p h.˛/ C 2D log 2 log p : The proof of Theorem 2.8 is given in Section B.3. Theorem 2.8 is trivial unless vp.˛/ D 0, in which case the factor .pD 1/ corresponds to the upper bound for the smallest positive integer s such that vp.˛s 1/ is positive. The dependence on p in Theorem 2.8 was slightly improved by Yamada [442] by means of a subtle variation of the proof of the main estimate of [129] for linear forms in two p-adic logarithms. Yamada’s result, reproduced as Theorem 12.3 and established in Chapter 12, has a striking application to an old conjecture of Erdős; see Theorem 7.11. The next result is a slight simplification of the estimate given on page 190 of Yu’s paper [449]. Theorem 2.9. Let p be a prime number and ˛1; : : : ; ˛n algebraic numbers in an algebraic number field of degree D. Let b1; : : : ; bn denote rational integers such that ˛b1 1 : : : ˛bn n is not equal to 1. Let A1; : : : ; An; B be real numbers with log Aj max n h.˛j /; 1 16e2D2 o ; 1 j n; and B maxfjb1j; : : : ; jbnj; 3g: If n 2, then we have vp.˛b1 1 : : : ˛bn n 1/.16eD/2.nC1/ n5=2 .log.2nD//2 Dn pD 1 .log p/2 log A1 : : : log An log B: Theorem 2.9 allows us to improve slightly the dependence on p in Theorem 2.8; see also Theorem 12.3. Note that (as in Theorems 2.10 and 2.11, but unlike in Theorem 2.12 and in [443]) we do not assume that vp.˛j / D 0 for j D 1; : : : ; n. The factor .pD 1/ in Theorem 2.9 and in Theorems 2.10 to 2.12 is ultimately due to the fact that the radius of convergence of the p-adic exponential function is finite, equal to p 1=.p 1/ . Removing it is a major open problem in the theory. This can be done under the rather restrictive assumption that vp.˛j 1/ 0 for j D 1; : : : ; n; see [168,169]. Theorem 2.9 should be compared with Theorem 2.2. We have exactly the same de- pendence on the parameters n, log A1; : : : ; log An, and B. A crucial point in Theorem 2.9 is the dependence on n, which is essential in the proofs of Theorems 8.2 and 8.3 towards
- 37. 2.3. Lower bounds for linear forms in p-adic logarithms 19 the abc-conjecture. These proofs also require an estimate for vp.˛b1 1 : : : ˛bn n 1/ with a (slightly) better dependence on p. For this reason, we quote the result stated on page 30 of [446]. Theorem 2.10. We keep the notation of Theorem 2.9. If n 1, then we have vp.˛b1 1 : : : ˛bn n 1/ 12 pD 1 log p 6.n C 1/D 2.nC1/ log.e5 nD/ max nh.˛1/ log p ; 1 o max nh.˛n/ log p ; 1 o log B: Yu [449] obtained an improvement of Theorem 2.10, with a better dependence on n, namely with nn in place of .n C 1/2n . We refer to [449] for a precise statement. The next statement, extracted from page 191 of Yu’s paper [449], is the p-adic analogue of the main result of [25]. Theorem 2.11. We keep the notation of Theorem 2.9. Let Bn be a real number such that B Bn jbnj: Assume that vp.bn/ vp.bj /; j D 1; : : : ; n: Let ı be a real number with 0 ı 1 2 . Then, we have vp.˛b1 1 : : : ˛bn n 1/ .16eD/2.nC1/ n3=2 .log.2nD//2 Dn pD 1 .log p/2 max n .log A1/ .log An/.log T /; ıB Bnc0.n; D/ o ; (2.8) where T D Bn ı c1.n; D/p.nC1/D .log A1/ .log An 1/ and c0.n; D/ D .2D/2nC1 log.2D/ log3 .3D/; c1.n; D/ D 2e.nC1/.6nC5/ D3n log.2D/: In particular, we get either B 2Bn.log A1/ .log An/ c0.n; D/ (2.9) or vp.˛b1 1 : : : ˛bn n 1/ .16eD/2.nC1/ n3=2 .log.2nD//2 Dn pD .log p/2 .log A1/ .log An/ max n 1; log c1.n; D/p.nC1/D B c0.n; D/ log An o : (2.10)
- 38. 20 2. Lower bounds for linear forms in complex and p-adic logarithms The formulation of Theorem 2.11 looks complicated, but it essentially corresponds to the introduction of the parameter B0 in Theorem 2.1 above. To get the second statement, we have selected ı0 D .log A1/ .log An/Bn B c0.n; D/ and distinguished the two cases ı0 1 2 , which gives (2.9), and ı0 1 2 , which, by (2.8) with ı D ı0, gives (2.10). In the special case bn D 1, the statements of Theorems 2.2 and 2.11 can be merged; see Theorem 3.2.8 of [183]. As for linear forms in complex logarithms, the case n D 2 is very important for applications. The next theorem reproduces one of the corollaries of the main result of [129]. Theorem 2.12. Let p be a prime number. Let ˛1 and ˛2 be multiplicatively independent algebraic numbers with vp.˛1/ D vp.˛2/ D 0. Set D D ŒQ.˛1; ˛2/ W Q. Let A1 and A2 be real numbers with log Aj max n h.˛j /; log p D o ; j D 1; 2: Let b1 and b2 be positive integers and set log B0 D max n log b1 D log A2 C b2 D log A1 C log log p C 0:4; 10 log p D ; 10 o : Then, we have the upper bound vp.˛b1 1 ˛b2 2 / 24p.pD 1/ .p 1/.log p/4 D4 .log A1/.log A2/ .log B0 /2 : (2.11) The assumption that ˛1 and ˛2 are multiplicatively independent is not really restrictive for the applications and is useful to get small numerical values in (2.11). The assumption that ˛1 and ˛2 are p-adic units is not restrictive. Indeed, if only one among vp.˛1/ and vp.˛2/ is positive, then vp.˛b1 1 ˛b2 2 / D 0, while, if both of them are positive, then vp.˛b1 1 ˛b2 2 / is bounded from below by a positive real number times the minimum of b1 and b2. A version of Theorem 2.12 (which is weaker only in terms of the numerical constants) is proved in this book; see Theorem 12.1. Unlike Theorem 2.9, Theorem 2.12 cannot be applied to bound vp.˛b 1/ from above because of the assumption of multiplicative independence, but see Theorem 12.3. Theorem 2.12 should be compared with Theorem 2.3. In particular, the numerical constant is very small and the quantity log B0 also occurs squared. We end this section with an improvement of Theorem 2.12 when ˛1 and ˛2 are rational numbers which are p-adically close to 1. Let x1 y1 and x2 y2 be non-zero rational numbers. Theorem 2.13 below, originated in [109], provides an explicit upper bound for the p-adic valuation of the difference between integer powers of x1 y1 and x2 y2 .
- 39. 2.3. Lower bounds for linear forms in p-adic logarithms 21 We suppose that there exist a positive integer g and a real number E with E 1C 1 p 1 and vp x1 y1 g 1 E; (2.12) and at least one of the two following conditions vp x2 y2 g 1 E (2.13) or vp x2 y2 g 1 1 and vp.b2/ vp.b1/ (2.14) is satisfied. For instance, if p is an odd prime number, then (2.12) and (2.13) hold with g D 1 if x1 y1 and x2 y2 are both congruent to 1 modulo p2 . We stress that the assumption “(2.13) or (2.14) holds” is more restrictive than the assumption “vp..x2 y2 /g 1/ 0” made in [109]. However, it seems that the latter assumption is not sufficient to derive Theorem 2 from Theorem 1 in [109], a fact which has been overlooked in [109]. Theorem 2.13. Let x1 y1 and x2 y2 be multiplicatively independent rational numbers. Let b1; b2 be positive integers. Assume that there exist a positive integer g and a real number E greater than 1 C 1 p 1 such that (2.12) holds, as well as (2.13) or (2.14). Let A1; A2 be real numbers with log Aj maxflog jxj j; log jyj j; E log pg; j D 1; 2; and put log B0 D max n log b1 D log A2 C b2 D log A1 C log.E log p/ C 0:4; 6 E log p; 5 o : Then, we have the upper bound vp x1 y1 b1 x2 y2 b2 36:1 g E3 .log p/4 .log A1/.log A2/.log B0 /2 ; (2.15) if p is odd or if p D 2 and v2.x2 y2 1/ 2. Observe that, if x1 y1 and x2 y2 are both congruent to 1 modulo p3 , then we can take E D 3 in Theorem 2.13 and the right hand side of (2.15) can be bounded independently of p. A weaker version of Theorem 2.13 is proved in this book; see Theorem 12.2. The parameter E in Theorem 2.13 should be compared with the parameter E in Theorems 2.1 and 2.4. Let us illustrate its importance with the following example taken from [117] (see also Exercise 6.11). Apply Theorem 2.13 with the prime number p to bound from above the p-adic valuation v of .x y /b .1 C cpk /, for positive integers k, b, c, x, y, M with k 3, 1 c pMk , x pk , and gcd.p; b/ D 1. The latter assumption implies that, if v 1, then the p-adic valuations of .x y /b 1 and x y 1 are positive
- 40. 22 2. Lower bounds for linear forms in complex and p-adic logarithms and equal. Consequently, pk divides x y 1 and we apply Theorem 2.13 with g D 1, A1 D p.MC1/k , A2 D maxfx; yg, and E D k to get v 36:1 log p log A1 E.log p/ .log A2/ .6 C log b/2 ; thus v 36:1.M C 1/ log p .log A2/ .6 C log b/2 : This upper bound is independent of k. It considerably improves the upper bound given by (2.11) and can be viewed as an analogue of Theorem 2.5 (with 1 replaced with M C1). 2.4. Notes F Linear forms in one logarithm have been studied by Mignotte and Waldschmidt [300]. Let ˛ be an algebraic number of degree D 2 and of small height. In their lower bound for j˛ 1j (that is, for j log ˛j), the dependence on D is much better than the one given by Liouville’s inequality Theorem B.10; see also [134]. Amoroso [7] proved that these results are essentially sharp; see also [171]. A p-adic analogue has been worked out in [105]. F In the homogeneous rational case, a complete proof of Theorem 2.1 up to the dependence on n of the numerical constant has been given by Waldschmidt in [433]. Nesterenko [316] has written a detailed proof of Theorem 2.2 in the special case where ˛1; : : : ; ˛n are rational numbers. Aleksentsev [5] established an estimate of a similar strength as Theorem 2.2. F There is a large gap between the size of the numerical constants in the best known lower bounds for linear forms in two and in three logarithms. For applications to Diophantine problems, it would be of greatest interest to improve the lower estimates for linear forms in three logarithms; see [298] for a step in this direction. F Bugeaud [111] obtained lower bounds for linear forms in two logarithms and simul- taneously for several p-adic valuations; see [52,54,111] for applications to Diophantine equations. F Simultaneous linear forms in logarithms are discussed in Chapter 9.
- 41. Chapter 3 First applications This chapter is principally devoted to Diophantine problems from which one can derive a linear form in complex logarithms in an almost obvious way, or, at least, in an easy way. We have listed them in increasing order of difficulty. We postpone to Chapter 4 applications of the theory of linear forms in complex logarithms to classical families of Diophantine equations, including unit equations, Thue equations, elliptic equations, etc. 3.1. On the distance between powers of 2 and powers of 3 One of the simplest applications of the theory of linear forms in complex logarithms shows that the distance between a power of 2 and the power of 3 closest to it tends to infinity when the power of 2 tends to infinity. In addition, it provides us with an explicit lower bound for this distance. Of course, and this will be clear in Section 3.3, the integers 2 and 3 can be replaced by any pair of multiplicatively independent positive integers. Theorem 3.1. For all positive integers m and n, we have j2m 3n j 2m .em/ 8:4108 : Proof. Let n 2 be an integer and define m and m0 by the conditions 2m0 3n 2m0 C1 and j3n 2m j D minf3n 2m0 ; 2m0 C1 3n g: Then, m is equal to m0 or to m0 C 1 and j2m 3n j 2m ; .m 1/ log 2 n log 3 .m C 1/ log 2: (3.1) The problem of finding a lower bound for j2m 3n j clearly reduces to this case. Also, we have m n. To show that 2m cannot be too close to 3n , it is sufficient to prove that the quantity ƒ WD 3n 2 m 1 is not too small in absolute value. Such a result is a direct consequence of Theorem 2.2, which, applied with ˛1 D 3; A1 D 3; ˛2 D 2; A2 D 2, gives that log jƒj 305 211=2 .log 2/ .log 3/ .log em/:
- 42. 24 3. First applications We conclude that j3n 2 m 1j .em/ 8:4108 ; and the theorem is established. As an application of Theorem 3.1, we list all the powers of 3 which differ from a power of 2 by at most 20. Our main auxiliary tool is the theory of continued fractions, briefly explained in Appendix A. Theorem 3.2. The only solutions in positive integers m; n with n 4 to the inequality j2m 3n j 20 (3.2) are given by j26 34 j D 17 and j28 35 j D 13. Proof. Let n 6 and m be integers satisfying (3.2). Applying Theorem 3.1 to Inequal- ity (3.2), we get 20 2m .em/ 8:4108 ; which implies log 20 m log 2 8:4 108 .1 C log m/; giving m 4 1010 and n .m C 1/.log 2/ log 3 3 1010 ; by (3.1). Since j log.1 C x/j 2jxj for any real number x with jxj 1 2 , the inequality j2m 3n j 20 implies ˇ ˇ ˇ ˇ log 3 log 2 m n ˇ ˇ ˇ ˇ 40 n log 2 3 n : (3.3) Observe that the right-hand side of (3.3) is less than 1 2n2 , since n 6. By Theorem A.2, the rational number m n is a convergent of the continued fraction expansion of WD log 3 log 2 . Furthermore, for n N WD 3 1010 , the smallest value of jm nj is obtained for the last convergent of the continued fraction expansion of with denominator less than N . The computation of this expansion shows that ˇ ˇ ˇ ˇ log 3 log 2 m n ˇ ˇ ˇ ˇ 9:6 10 18 ; for 0 n 3 1010 : Comparing (3.3) with this estimate, we deduce that n 36. A rapid verification in the range 6 n 36 completes the proof that (3.2) has no solution with n 6.
- 43. 3.2. Effective irrationality measures for quotients of logarithms of integers 25 3.2. Effective irrationality measures for quotients of logarithms of integers Let a and b be multiplicatively independent positive rational numbers. It follows straight- forwardly from the first assertion of Theorem 2.1 that the real number log a log b is transcendental and, by arguing as in the proof of Theorem 3.1, we show that it is not a Liouville number (see Definition A.6) and give an effective upper bound for its irrationality exponent (see Definition A.3). Theorem 3.3. Let a1; a2; b1; b2 be positive integers such that the rational numbers a1 a2 and b1 b2 are multiplicatively independent. Then, the effective irrationality exponent of the real number .log a1 a2 /=.log b1 b2 / satisfies eff log a1 a2 log b1 b2 .log maxfa1; a2g/ .log maxfb1; b2g/: A strengthening of Theorem 3.3 in the special case where a1 a2 and b1 b2 are rational numbers close to 1 is given in Section 5.1. Proof. Let p q be a convergent to .log a1 a2 /=.log b1 b2 / with q 2. Since an irrational real number and its inverse have the same irrationality exponent, we can assume that j log a1 a2 j j log b1 b2 j. It then follows from Theorem 2.1 (or from Theorem 2.2) that log ˇ ˇ ˇ ˇq log a1 a2 p log b1 b2 ˇ ˇ ˇ ˇ .log maxfa1; a2g/ .log maxfb1; b2g/ .log q/: Thus, there exists an effectively computable, absolute real number C such that ˇ ˇ ˇ ˇ log a1 a2 log b1 b2 p q ˇ ˇ ˇ ˇ D jq log a1 a2 p log b1 b2 j q log b1 b2 q C.log maxfa1;a2g/ .log maxfb1;b2g/ : This proves the theorem. The fact that the dependence on B in Theorem 2.2 occurs through the factor .log B/ and not through the factor .log B/2 , as in Theorem 2.3, is crucial for the proof of Theorem 3.3. 3.3. On the distance between two integral S-units For a finite set S of prime numbers, an integral S-unit is, by definition, an integer all of whose prime factors are in S. The following result extends Theorem 3.1. It was proved by Tijdeman [414] in 1973. Theorem 3.4. Let S denote a finite, non-empty set of prime numbers and .xj /j 1 the increasing sequence of all positive integers whose prime factors belong to S. There exists an effectively computable real number C, depending only on S, such that xj C1 xj xj .log 2xj / C ; j 1:
- 44. 26 3. First applications Proof. Write S D fq1; : : : ; qsg, where q1; : : : ; qs are prime numbers and q1 qs. Let u; v with 3 u v be two consecutive elements in the sequence .xj /j 1 and write u D s Y iD1 qui i ; v D s Y iD1 qvi i : Without any loss of generality, we can assume that s 2, v 2u, and that u and v are coprime. Thus, for every i D 1; : : : ; s, at least one of ui ; vi is zero. Set ƒ WD s Y iD1 qvi ui i 1 D v u 1: Put B D maxf3; jv1 u1j; : : : ; jvs usjg and observe that 2B=3 qB=3 1 v 2u: (3.4) It follows from Theorem 2.2 that log ƒ 30sC4 s5 s Y iD1 log qi .log eB/: Combined with (3.4), this gives log v u 1 30sC4 s5 s Y iD1 log qi log 3e log.2u/ log 2 : Taking the exponential of both sides and multiplying by u, we get the assertion of the theorem. In view of the next result, established by Tijdeman [415] in 1974, Theorem 3.4 is essentially best possible. Theorem 3.5. Let P be an integer and S a finite set of at least two distinct prime numbers all being at most equal to P . Let .xj /j 1 be the increasing sequence of all positive integers whose prime divisors belong to S. There exists an effectively computable real number C, depending only on P , such that xj C1 xj xj .log 2xj / C ; j 1: It is an open problem of Erdős to establish that, for S being the set of all prime numbers less than P , Theorem 3.5 holds with a real number C D C.P / which tends to infinity with P . 3.4. Effective irrationality measures for n-th roots of algebraic numbers We have seen in Theorem 3.3 how the theory of linear forms in logarithms can be applied to give an effective upper bound for the irrationality exponent of certain transcendental real numbers. The next result addresses a special class of real algebraic numbers.
- 45. 3.5. On the greatest prime factor of values of integer polynomials 27 Theorem 3.6. Let n 3 be an integer and a positive, real algebraic number. Then, the effective irrationality exponent of n p satisfies eff. n p / log n: Furthermore, if a; b are integers with 1 b a, then we have eff. n p a=b / .log a/.log n/; and, if a 2n , then we get eff. n p a=b / 11000 log a: (3.5) When n is sufficiently large in terms of , Theorem 3.6 improves the upper bound eff. n p / n deg./, given by Liouville’s Theorem A.5. It remains, however, very far from Roth’s Theorem A.7, which asserts that . n p / D 2 but gives no information on eff. n p /. In Section 5.2 we considerably strengthen Theorem 3.6 when is a rational number very close to 1. Proof. Let n 3 be an integer and a positive, real algebraic number. Let p q be a convergent of n p with q 2. Observe that ˇ ˇ ˇ n p p q ˇ ˇ ˇ 1 2n maxf1; jjg ˇ ˇ ˇ p q nˇ ˇ ˇ: It directly follows from Theorem 2.2 that log ˇ ˇ ˇ p q nˇ ˇ ˇ C./.log q/.log n/; where C./ is an effectively computable real number which can be expressed in terms of the height and the degree of . If is the rational number a b with 1 b a, then C./ can be taken to be an absolute real number times log a. This proves the first two statements of the theorem. For large values of n, we obtain a stronger result (not only numerically) by applying Theorem 2.3. Indeed, if a; b are integers with 1 b a, it directly gives the lower bound log ˇ ˇ ˇn log p q log a b ˇ ˇ ˇ 25:2.log p/.log a/ max n log 2n log a C 0:21; 20 o2 ; if p is sufficiently large. From this and under the assumption a 2n , we deduce (3.5). 3.5. On the greatest prime factor of values of integer polynomials Let f .X/ be a non-zero integer polynomial with at least two distinct roots. Let n be an integer. Using the theory of linear forms in logarithms, Shorey and Tijdeman [374] were the first to give an effective lower bound for the greatest prime factor of f .n/ as n tends
- 46. 28 3. First applications to infinity. In 1998 Tijdeman [418] noticed that, for the polynomial X.X C 1/, the use of Theorem 2.2 allows us to improve the results of [374]. In his doctoral thesis, Haristoy [221] extended Tijdeman’s result to an arbitrary polynomial f .X/. For an integer n, let denote by P Œn its greatest prime factor with the convention that P Œ0 D P Œ˙1 D 1. Theorem 3.7. Let f .X/ be an integer polynomial with at least two distinct roots. Then, we have P Œf .n/ f log log n log log log n log log log log n ; for n 107 : (3.6) A result similar to Theorem 3.7 has been established in [217]; see also Chapter 8. If we apply Theorem 2.1 (with a weaker dependence on the number of logarithms in the linear form than in Theorem 2.2) instead of Theorem 2.2, then we would get the weaker lower bound P Œf .n/ f log log n; for n 100; which was obtained in [374]. Proof. To avoid technical complications, we treat only the case of the polynomial X.X C1/. For an integer n at least equal to 100, write n.n C 1/ D pu1 1 : : : puk k ; where p1 p2 denotes the sequence of prime numbers in increasing order and the integers u1; : : : ; uk are non-negative. There exist disjoint non-empty subsets I and J of f1; : : : ; kg such that I [ J D f1; : : : ; kg and n D Y i2I pui i ; n C 1 D Y j 2J p uj j : Then, there exist 1; : : : ; k in f˙1g such that n C 1 n 1 D jp1u1 1 : : : pk uk k 1j D 1 n : (3.7) Observe that n C 1 2uj ; j D 1; : : : ; k; thus max 1j k uj 2 log n: (3.8) Since pj j log j , for j D 2; : : : ; k, by Theorem D.2, we deduce from (3.8) and Theorems 2.2 and D.2 that there exist effectively computable, absolute real numbers c1; c2; c3 such that log jp1u1 1 : : : pk uk k 1j ck 1 .log p1/ .log pk/ .log log n/ c k log log k 2 .log log n/; thus, by (3.7), log n c k log log k 3 log log n :
- 47. 3.5. On the greatest prime factor of values of integer polynomials 29 By Theorem D.2, this implies that pk k log k log log n log log log n log log log log n ; as asserted. The general case of an arbitrary monic, integer polynomial with distinct roots ˛ and ˇ rests on the same idea, but is technically more complicated. Let OK denote the ring of integers of the algebraic number field K WD Q.˛; ˇ/ and h its class number. Let n be an integer greater than j˛j and jˇj. Write the integer ideal .n ˛/.n ˇ/OK as a product of powers of prime ideals and raise this equality to the power h. Then, the quantity .n ˛/h .n ˇ/ h , which is equal to 1 C O.1 n /, can, with the help of Proposition C.5, be expressed as a product of powers of algebraic numbers in K whose heights are controlled. It then follows from Theorem 2.2 that this product cannot be too close to 1, hence the result. We leave the details to the reader. Let S be a finite non-empty set of prime numbers. For a non-zero integer n, write ŒnS WD Y p2S jnj 1 p ; where j jp is the p-adic absolute value normalized such that jpj 1 p D p. Said differently, ŒnS is the greatest divisor of n, all of whose prime factors belong to S. In 1984, Stewart [396] applied the theory of linear forms in logarithms to prove non- trivial effective upper bounds for Œn.n C 1/ : : : .n C k/S , for any positive integer k. His result was later extended by Gross and Vincent [205] as follows; see also [122]. Theorem 3.8. Let f .X/ be an integer polynomial with at least two distinct roots and S a finite, non-empty set of prime numbers. There exist effectively computable positive real numbers c1 and c2, depending only on f .X/ and S, such that, for every non-zero integer n which is not a root of f .X/, we have Œf .n/S c1jf .n/j1 c2 : Proof. To avoid technical complications, we treat only the case of the polynomial X.X C1/. We leave to the reader the details of the general case. Let n be an integer with jnj 2. Write S D fq1; : : : ; qsg, where s is the cardinality of S and q1 qs. Let u1; : : : ; us be non-negative integers and a an integer coprime with q1 qs such that n.n C 1/ D qu1 1 : : : qus s a: There exist disjoint non-empty subsets I and J of f1; : : : ; sg such that I [J D f1; : : : ; sg and integers a0 ; a00 such that a D a0 a00 and n D a0 Y i2I qui i ; n C 1 D a00 Y j2J q uj j : Then, there exist 1; : : : ; s in f˙1g such that 1 n D n C 1 n 1 D q1u1 1 : : : qsus s a00 a0 1 DW ƒ:
- 48. 30 3. First applications Noticing that uj 2 log jnj for j D 1; : : : ; s, we deduce from Theorem 2.2 applied to ƒ with the quantity B00 that there exists an effectively computable real number c3 such that log jnj cs 3.log q1/ .log qs/.log maxfja0 j; ja00 j; 2g/ log 2.log jnj/.log qs/ log maxfja0j; ja00j; 2g : This shows that there exists an effectively computable positive real number c4, depending only on the set S, such that maxfja0 j; ja00 j; 2g jnjc4 : If jnj 21=c4 , then we get a maxfja0 j; ja00 j; 2g jnjc4 and Œn.n C 1/S D qu1 1 : : : qus s D n.n C 1/ a n.n C 1/ jnjc4 .n.n C 1//1 c4=3 ; which gives the desired result. We now explain how a suitable version of Theorem 3.8 with explicit values for c1 and c2 implies Theorem 3.7. Observe that, setting C WD cs 3.log q1/ .log qs/, an admissible value for c4 is given by c4 D c5C 1 .log.C log qs// 1 ; for some suitable absolute real number c5. If S is the set composed of the first s prime numbers, then it follows from Theorem D.2 that c4 exceeds c s log log s 6 for some absolute real number c6. If P Œn.n C 1/ D ps, then maxfja0 j; ja00 j; 2g D 2 and jnjc4 2, thus, log jnj 2c s log log s 6 : Combined with Theorem D.2, this gives P Œn.n C 1/ log log n log log log n log log log log n ; and we recover Theorem 3.7. 3.6. On the greatest prime factor of terms of linear recurrence sequences Let k be a positive integer, a1; : : : ; ak and u0; : : : ; uk 1 be integers such that ak is non-zero and u0; : : : ; uk 1 are not all zero. Put un D a1un 1 C C akun k; for n k: The sequence .un/n0 is a linear recurrence sequence of integers of order k. Its companion (or, characteristic) polynomial G.z/ WD zk a1zk 1 ak
- 49. 3.6. On the greatest prime factor of terms of linear recurrence sequences 31 factors as G.z/ D t Y iD1 .z ˛i /`i ; where ˛1; : : : ; ˛t are distinct algebraic numbers, ordered in such a way that j˛1j j˛t j, and `1; : : : ; `t are positive integers. Then, (see e.g. [179] or Chapter C of [376]) there exist polynomials f1.X/; : : : ; ft .X/ of degrees less than `1; : : : ; `t , respectively, and with coefficients in Q.˛1; : : : ; ˛t /, such that un D f1.n/˛n 1 C C ft .n/˛n t ; for n 0: The recurrence sequence .un/n0 is said to be degenerate if there are integers i; j with 1 i j t such that ˛i ˛j is a root of unity. We say that .un/n0 has a dominant root if j˛1j j˛2j and f1.X/ is not the zero polynomial. The next result was established in [399]; see also [395]. Theorem 3.9. Let u WD .un/n0 be a recurrence sequence of integers given by un D f1.n/˛n 1 C C ft .n/˛n t ; for n 0; and having a dominant root ˛1. For any integer n greater than 100 and such that un is not equal to f1.n/˛n 1 , the greatest prime factor of un satisfies P Œun u log n log log n log log log n : (3.9) Proof. We establish a slightly more general result. We consider a sequence of non-zero integers v WD .vn/n0 with the property that there are in .0; 1/ and a positive real number C such that jvn f .n/˛n j C j˛jn ; n 0; where f .X/ is a non-zero polynomial whose coefficients are algebraic numbers and ˛ is an algebraic number with j˛j 1. Clearly, a recurrence sequence having a dominant root ˛ has the above property. Let p1; p2; : : : be the sequence of prime numbers in increasing order. Let n be a positive integer such that f .n/ is non-zero and vn ¤ f .n/˛n . Assume that the greatest prime factor of vn is equal to pk, that is, assume that vn D pr1 1 prk k ; where r1; : : : ; rk are non-negative integers and rk 1. Then, we have ƒ WD jpr1 1 prk k f .n/ 1 ˛ n 1j Cjf .n/j 1 j˛j. 1/n ; and, since 1, we get log ƒ v . n/: As vn ¤ f .n/˛n , the quantity ƒ is non-zero and Theorem 2.2 implies that log ƒ v ck 1 .log p1/ .log pk/ .log n/2 ;
- 50. 32 3. First applications where c1 is an effectively computable, positive, absolute real number. Comparing both estimates for log ƒ, we obtain that n v .c1 log pk/k .log n/2 ; thus, by Theorem D.2, log n v pk log pk log log pk: This implies that P Œvn D pk v log n log log n log log log n ; and the theorem is established. The lower bound for P Œun in (3.9) is of similar strength as the one for P Œf .n/ in Theorem 3.7, since junj grows exponentially fast in n. Both proofs show that f in (3.6) and u in (3.9) can be replaced by .1 /, for any positive and any sufficiently large integer n. 3.7. Perfect powers in linear recurrence sequences We continue the study of arithmetical properties of linear recurrence sequences of integers having a dominant root and show that no term of such a sequence can be a very large power of an integer greater than one. Theorem 3.10. Let u WD .un/n0 be a recurrence sequence of integers given by un D f1.n/˛n 1 C C ft .n/˛n t ; for n 0; and having a dominant root ˛1. Assume that ˛1 is a simple root, that is, the polynomial f1.X/ is equal to a non-zero algebraic number f1. Then, the equation un D yq ; in integers n; y; q with q a prime number, jyj 2, and un ¤ f1˛n 1 , implies that q u 1: Theorem 3.10 was proved in [372]; see also [190,325]. More can be said in the case of binary recurrence sequences, see Section 6.9. Proof. We establish a slightly more general result. We consider a sequence of non-zero integers v WD .vn/n0 with the property that there are in .0; 1/ and a positive real number C such that jvn f ˛n j C j˛jn ; n 0; where f is a non-zero algebraic number and ˛ is an algebraic number with j˛j 1. Let n; y; q with q a prime number be such that vn D yq , jyj 2, and vn ¤ f ˛n . Then, we have jf 1 ˛ n yq 1j C jf j j˛j. 1/n : (3.10)
- 51. 3.7. Perfect powers in linear recurrence sequences 33 There exist integers k and r such that n D kq C r with jrj q 2 . Rewriting (3.10) as ƒ WD ˇ ˇ ˇf 1 y ˛k q ˛ r 1 ˇ ˇ ˇ C jf j j˛j. 1/n ; we observe that log ƒ v . n/: (3.11) By Theorem B.5, the height of ˛k y is bounded from above by the sum of log jyj and k times the height of ˛. Since q log jyj v n .k C 1/q and kq 2 n v q log jyj; (3.12) we get that h.˛k =y/ v log jyj. It then follows from Theorem 2.2 that log ƒ v .log jyj/.log q/; which, combined with (3.11) and (3.12), gives q log jyj v n v .log jyj/.log q/; and we get q v 1. This proves the theorem. We point out a very special case of Theorem 3.10. Corollary 3.11. Let ˛ and ˇ be algebraic integers such that ˛ C ˇ, ˛ˇ are rational integers and ˛ jˇj. Then, the equation ˛n ˇn ˛ ˇ D yq .resp., ˛n C ˇn D yq /; (3.13) in integers n; y; q with jyj 2 and q a prime number, implies that q ˛;ˇ 1: Let us discuss a celebrated example of resolution of equations of the form (3.13), namely the determination of all the perfect powers among the Fibonacci and Lucas num- bers. Set D 1C p 5 2 and observe that 1 D 1 p 5 2 . Let the integer sequences .Fn/n0 and .Ln/n0 be defined respectively by Fn D n . 1 /n . 1/ ; Ln D n C . 1 /n ; so that the first elements of the Fibonacci sequence .Fn/n0 and of the Lucas sequence .Ln/n0 are given by 0; 1; 1; 2; 3; 5; 8; 13; 21; 34; 55; 89; 144; 233; : : : and 2; 1; 3; 4; 7; 11; 18; 29; 47; 76; : : : ; respectively. The Fibonacci sequence is an emblematic example of a Lucas–Lehmer sequence (see Section 7.2). The following result has been established in [137].
- 52. Exploring the Variety of Random Documents with Different Content
- 53. CHAPTER VI. ATTRACTION OF COHESION. In previous chapters one kind of attraction—viz., that of gravitation, has been discussed and illustrated in a popular manner, and pursuing the examination of the invisible, active, and real forces of nature, the attraction of cohesion will next engage our attention. There is a peculiar satisfaction in pursuing such investigations, because every step is attended by a reasonable proof; there is no ghostly mystery in philosophic studies; the mind is not suddenly startled at one moment with that which seems more than natural; it is not carried away in an ecstasy of wonder and awe, as in the so- called spirit-rapping experiments, to be again rudely brought back to the material by the disclosure of trickeries of the most ludicrous kind, such as those lately exposed by M. Jobert de Lamballe, at the Academy of Sciences at Paris. This gentleman has unmasked the effrontery of the spirit-rappers by merely stripping the stocking from the heel of a young girl of fourteen. M. Velpeau declares that the rapping is produced by the muscles of the heel and knee acting in concert, and quotes the case of a lady once celebrated as a medium, who has the power of producing the most curious and interesting music with the tendons of the thigh. This music is said to be loud enough to be heard from one end of a long room to the other, and has often played a conspicuous part in the revelations made by the medium. M. Jules Clocquet also explained the method by which the famous girl pendulum had so long abused the credulity of the Paris public. This girl, whose self-styled faculty is that of striking the hour at any time of the day or night, was attended at the Hospital St. Louis by M. Clocquet, who states that the vibrations in this case were produced by a rotatory motion in the lumbar regions of the vertebral column. The sound of these (à la rattlesnake) was so powerful, that they might be distinctly heard at a distance of twenty-five feet. In studying the powers of nature, which the most sceptical mind allows must exist, there is an abundant field for experiment without attempting the exploits of Macbeth's witches, or the fanciful powers of Manfred; and, returning to the theme of our present chapter, it may be asked, how is cohesion defined? and the answer may be given, by directing attention to the three physical conditions of water, which assumes the form of ice, water, or steam. In the Polar regions, and also in the Alpine and other mountains where glaciers exist, there the traveller speaks of ice twenty, thirty, forty, nay, three hundred feet in thickness. Here the withdrawal of a certain quantity of heat from the water evidently allows a new force to come into full play. We may call it what we like; but cohesion, from the Latin cum, together, and hæreo, I stick or cleave, appears to be the best and most rational term for this power which tends to make the atoms or particles of the same kind of matter move towards each other, and to prevent them being separated or moved asunder. That it is not merely hypothetical is shown by the following experiments. If two pieces of lead are cast, and the ends nicely scraped, taking care not to touch the surfaces with the fingers, they may by simple pressure be made to cohere, and in that state of attraction may be lifted from the table by the ring which is usually inserted for convenience in the upper piece of lead; they may be hung for some time from a proper support, and the lower bit of lead will not break away from the upper one; they may even be suspended, as demonstrated by Morveau, in the vacuum of an air-pump, to show that the cohesion is not mistaken for the pressure of the atmosphere, and no separation occurs. And when the union is broken by physical force, it is surprising to notice the limited number of points, like pin points, where the cohesion has occurred; whilst the weight of the lump of lead upheld against the force of gravitation reminds one forcibly of the attraction of a mass of soft iron by a powerful magnet, and leads the philosophic inquirer to speculate on the principle of cohesion being only some masked form of magnetic or electrical attraction. (Fig. 75.)
- 54. Fig. 75. a a. Two pieces of lead, scraped clean at the surfaces b b. c. Stand, supporting the two pieces of lead attached to each other by cohesion. A fine example of the same force is shown in the use of a pair of flat iron surfaces, planed by the celebrated Whitworth, of Manchester. These surfaces are so true, that when placed upon each other, the upper one will freely rotate when pushed round, in consequence of the thin film of air remaining between the surfaces, which acts like a cushion, and prevents the metallic cohesion. When, however, the upper plate is slid over the lower one gradually, so as to exclude the air, then the two may be lifted together, because cohesion has taken place. (Fig. 76.) Fig. 76. a. Whitworth's planes, with film of air between them. b. Film of air excluded when cohesion occurs. A glass vessel is a good example of cohesion. The materials of which it is composed have been soft and liquid when melted in the fire, and on the removal of the excess of heat it has become hard and solid, in consequence of the attractive force of cohesion binding the particles together; in the absence of such a power, of course, the material would fall into the condition of dust, and a mere shapeless heap of silicates of potash and lead would indicate the place where the moulded and coherent glass would otherwise stand.
- 55. A lump of lead, six inches long by four broad, and half an inch thick, may be supported by dexterously taking off a thick shaving with a proper plane, and after pressing an inch or more of the strip on the planed surface of the large lump of lead, the cohesion is so powerful that the latter may be lifted from the table by the strip of metal. The bullets projected from Perkins' steam-gun, at the rate of three hundred per minute, are thrown with such violence, that, when received on a thick plate of lead backed up with sheet iron, a cold welding takes place between the two surfaces of metal in the most perfect manner, just as two soft pieces of the metal potassium may be squeezed and welded together. The surfaces of an apple torn asunder will not readily cohere, but if cut with a sharp knife, cohesion easily occurs; so with a wound produced by a jagged surface, it is difficult to make the parts heal, whereas some of the most desperate sabre-cuts have been healed, the cohesion of the surfaces of cut flesh being very rapid; hence, if the top of a finger is cut off, it may be replaced, and will grow, in consequence of the natural cohesion of the parts. The art of plating copper with silver, which is afterwards gilt, and then drawn out into flattened wire for the manufacture of gold lace and epaulets, usually termed bullion, is another example of the wonderful cohesion of the particles of gold, of which a single grain may be extended over the finest plate wire measuring 345 feet in length. The process of making wax candles is a good illustration of the attraction of cohesion; they are not generally cast in moulds, as most persons suppose, but are made by the successive applications of melted wax around the central plaited wick. Other examples of cohesion are shown by icicles, and also stalagmites; which latter are produced by the gradual dropping of water containing chalk (carbonate of lime) held in solution by the excess of carbonic acid gas; the solvent gradually evaporates, and leaves a series of calcareous films, and these cohere in succession, producing the most fantastic forms, as shown in various remarkable caverns, and especially in the cave of Arta, in the island of Majorca. In metallic substances the cohesion of the particles assumes an important bearing in the question of relative toughness and power of resisting a strain; hence the term cohesion is modified into that of the property of tenacity. The tenacity of the different metals is determined by ascertaining the weight required to break wires of the same length and gauge. Iron appears to possess the property of tenacity in the greatest, and lead in the least degree. (Fig. 77.)
- 56. Fig. 77. b. Pan supported by leaden wire broken by a weight which the iron wire at a easily supports. The tenacity of iron is taken advantage of in the most scientific manner by the great engineers who have constructed the Britannia Tube, and that eighth wonder of the world, the Leviathan, or Great Eastern steam-ship. In both of these sublime embodiments of the genius and industrial skill of Great Britain the advantage of the cellular principle is fully recognised. The magnitude of this colossal ship is better realized when it is remembered that the Great Eastern is six times the size of the Duke of Wellington line-of-battle ship, that her length is more than three times that of the height of the Monument, while in breadth it is equal to the width of Pall Mall, and that a promenade round the deck will afford a walk of more than a quarter of a mile. Up to the water-mark the hull is constructed with an inner and outer shell, two feet ten inches apart, each of three-quarter-inch plate; and between them, at intervals of six feet, run horizontal webs of iron plates, which convert the whole into a series of continuous cells or iron boxes. (Fig. 78.) Fig. 78. Transverse section of Great Eastern, showing the cellular construction from keel to water-line, a a. This double ship is useful in various ways; in the first place, the danger arising from collision is diminished, as it is supposed that the outer web only would be broken through or damaged; so that
- 57. the water would not then rush into the steam-ship, but merely fill the space between the shells. In the second place, if there should be any difficulty in procuring ballast, the space can be filled with 2500 tons of water, or again pumped out, according to the requirements of the vessel. The strength of a continued cellular construction can be easily imagined, and may be well illustrated by a thin sheet of common tin plate. If the ends be rested on blocks of wood, so as to lap over the wood about one inch, they are easily displaced, and the mimic bridge broken down from its supports by the addition to the centre of a few ounce weights; whilst the same tin plate rolled up in the figure of a tube, and again rested on the same blocks, will now support many pounds weight without bending or breaking down. (Fig. 79.) Fig. 79. a. Flat tin plate, breaking down with a few ounce weights. b. Same tin plate rolled up supports a very heavy weight. The deck of the ship is double or cellular, after the plan of Stephenson in the Britannia Tubular Bridge, and is 692 feet in length. The tonnage register is 18,200 tons, and 22,500 tons builder's measure; the hull of the Great Eastern is considered to be of such enormous tenacity, that, if it were supported by massive blocks of stone six feet square, placed at each end, at stem and stern, it would not deflect, curve, or bend down in the middle more than six inches even with all her machinery, coals, cargo, and living freight. In adducing remarkable instances of the adhesive power and tenacity of inorganic matter, it may not be altogether out of place to allude to the strength and force of living matter, or muscular power. It is stated that Dr. George B. Winship, of Roxbury in America, a young physician, twenty-five years old, and weighing 143 pounds, is the strongest man alive; in fact, quite the Samson of the nineteenth century. He can raise a barrel of flour from the floor to his shoulders; can raise himself with either little finger till his chin is half a foot above it; can raise 200 pounds with either little finger; can put up a church bell of 141 pounds; can lift with his hands 926 pounds dead weight without the aid of straps or belts of any kind. As compared with Topham, the Cornish strong man, who could raise 800 pounds, or the Belgic one, his power is greater; and as the use of straps and belts increases the power of lifting by about four times, it is stated that Winship could lift at least 2500 pounds weight. With these illustrations of cohesion we may return again to the abstract consideration of this power with reference to water, in which we have noticed that the antagonist to this kind of attraction is the force or power termed caloric or heat. The latter influence removes the frozen bands of winter and converts the ice to the next condition, water. In this state cohesion is almost concealed, although there is just a slight excess to hold even the particles of water in a state of unity, and this fact is beautifully illustrated by the formation of the brilliant diamond drops of dew on the surfaces of various leaves, as also in the force and power exercised by great volumes of water, which exert their mighty strength in the shape of breaker-waves, dashing against rocks and lighthouses, and making them tremble to their very base by the violence of the shock; here there must be some unity of particles, or the collective strength could not be exerted, it would be like throwing a handful of sand against a
- 58. Fig. 80. a. Ordinary glass water hammer. b. Copper tube ditto, showing exhausting syringe at d, the height of the water at b, and the end to be placed in the fire at c. window—a certain amount of noise is produced, but the glass is not fractured; whilst the same sand united by any glutinous material, would break its way through, and soon fracture the brittle glass. It is so usual to see the particles of water easily separated, that it becomes difficult to recognise the presence of cohesion; but this bond of union is well illustrated in the experiment of the water hammer. The little instrument is generally made of a glass tube with a bulb at one end; in this bulb the water which it contains is boiled, and as the steam issues from the other extremity, drawn out to a capillary tube, the opening is closed by fusion with the heat of a blowpipe flame. As the water cools the steam condenses, and a vacuum, so far as air is concerned, is produced; if now the tube is suddenly inverted, the whole of the water falls en masse, collectively, and striking against the bottom of the tube, produces a metallic ring, just as if a piece of wood or metal were contained within the tube. If the end to which the water falls is not well cushioned by the palm of the hand, the water hammers itself through and breaks away that part of the glass tube. Hence it is better to construct the water hammer of copper tube, about three-quarters of an inch in diameter and three feet long; at one end a female screw-piece is inserted, into which a stop-cock is fitted; when the tube is filled to the height of about six inches with water, and shaken, the air divides the descending volume of water, and the ordinary splashing sound is heard; there is no unity or cohesion of the parts; if, however, the end of the copper tube is thrust into a fire and the water boiled so that steam issues from the cock, which is then closed, and the tube removed and cooled, a smart blow is given, and distinctly heard when the copper tube is rapidly inverted or shaken so as to cause the water to rise and fall. The experiment may be rendered still more instructive by turning the cock and admitting the air, which rushes in with a whizzing sound, and on shaking the tube the metallic ring is no longer heard, but it may be again restored by attaching a small air syringe or hand pump, and removing the air by exhaustion. (Fig. 80.) In the fluid condition water still possesses a surplus of cohesion over the antagonistic force of heat; when, however, the latter is applied in excess, then the quasi-struggle terminates; the heat overpowers the cohesive attraction, and converts the water into the most willing slave which has ever lent itself to the caprices of man—viz., into steam—glorious, useful steam: and now the other end of the chain is reached, where heat triumphs; whilst in solids, such as ice, cohesion is the conqueror, and the intermediate link is displayed in the fluid state of water. If any fact could give an idea of the gigantic size of the Great Eastern, it is the force of the steam which will be employed to move it at the rate of about eighteen miles per hour with a power estimated at the nominal rate of 2600 horses, but absolutely of at least 12,000 horses. This steam power, coupled with the fact that she has been enormously strengthened in her sharp, powerful bows, by laying down three complete iron decks forward, extending from the bows backward for 120 feet, will demonstrate that in case of war the Great Eastern may prove to be a powerful auxiliary to the Government. These decks will be occupied by the crew of 300 or 400 men, and with this large increase of strength forward, the Great Eastern, steaming full power, could overtake and cut in two the largest wooden line-of-battle ship that ever floated. Should war unhappily spread to peaceful England, and the enormous power of this vessel be realized, her name would not inappropriately be changed from the Great Eastern to the Great Terror of the ocean. The Times very properly inquires, What fleet could stand in the way of such a mass, weighing some 30,000 tons, and driven through the water by 12,000 horse-power, at the rate of twenty-two or twenty-three miles per hour. To produce the steam, 250 tons of coal per diem will be required, and great will be the
- 59. honourable pride of the projectors when they see her fairly afloat, and gliding through the ocean to the Far West. A good and striking experiment, displaying the change from the liquid to the vapour state, is shown by tying a piece of sheet caoutchouc over a tin vessel containing an ounce or two of water. When this boils, the india-rubber is distended, and breaks with a loud noise; or in another illustration, by pouring some ether through a funnel carefully into a flask placed in a ring stand. If flame is applied to the orifice, no vapour issues that will ignite, provided the neck of the flask has not been wetted with the ether. When, however, the heat of a spirit-lamp is applied, the ether soon boils, and now on the application of a lighted taper, a flame some feet in length is produced, which is regulated by the spirit- lamp below, and when this is removed, the length of the flame diminishes immediately, and is totally extinguished if the bottom of the flask is plunged into cold water; the withdrawal of the heat restores the power of cohesion. Another illustration of the vast power of steam will be shortly displayed in the Steam Ram; and, Supposing, says the Times, the new steam ram to prove a successful design, the finest specimens of modern men-of-war will be reduced by comparison to the helplessness of cock boats. Conceive a monstrous fabric floating in mid-channel, fire proof and ball proof, capable of hurling broadsides of 100 shot to a distance of six miles; or of clapping on steam at pleasure and running down everything on the surface of the sea with a momentum utterly irresistible. This terrible engine of destruction is expected to be itself indestructible. We are told that she may be riddled with shot (supposing any shot could pierce her sides), that she may have her stem and her stern cut to pieces, and be reduced apparently to a shapeless wreck, without losing her buoyancy or power. Supposing that she relies upon the shock of her impact instead of fighting her guns, it is calculated that she would sink a line-of-battle ship in three minutes, so that a squadron as large as our whole fleet now in commission would be destroyed in about one hour and a quarter.
- 60. CHAPTER VII. ADHESIVE ATTRACTION. The term cohesion must not be confounded with that of adhesion, which refers to the clinging to or attraction of bodies of a dissimilar kind. The late Professor Daniell defines cohesion to be an attraction of homogeneous (ὁμος, like, and γενος, kind) or similar particles; adhesion to be an attraction subsisting between particles of a heterogeneous, ετερος, different, and γενος, kind. There are numerous illustrations of adhesion, such as mending china, and the use of glue, or paste, in uniting different surfaces, or mortar, in building with bricks; it is also well shown at the lecture table by means of a pair of scales, one scale-pan of which being well cleaned with alkali at the bottom, may then be rested on the surface of water contained in a plate; the adhesion between the water and the metal is so perfect, that many grain weights may be placed in the other pan before the adhesion is broken; and after breakage, if the pan be again placed on the water, and a few grains removed from the other, so as to adjust the two pans, and make them nearly equal, a drop of oil of turpentine being added, instantly spreads itself over the water, and breaking the adhesion between the latter and the metal, the scale-pan is immediately and again broken away, as the adhesion between the turpentine and the metal is not so great as that of water and metal. The adhesion of air and water is well displayed in an apparatus recommended for ventilating mines, in which a constant descending stream of water carries with it a quantity of air, which being disengaged, is then forced out of a proper orifice. The same kind of adhesion between air and water is displayed in the ancient Spanish Catalan forge, where the blast is supplied to the iron furnace on a similar principle, only, a natural cascade is taken advantage of instead of an artificial fall of water through a pipe. The adhesion of air and water becomes of some value when a river flows through a large and crowded city, because the water in its passage to and fro, must necessarily drag with it, a continuous column of air, and assist in maintaining that constant agitation of the air which is desirable as a preventive to any accumulation of noxious air charged with fœtid odours, arising from mud banks or from other causes. The fact of adhesion, existing between water and air, is readily shown, by resting one end of a long glass tube, of at least one inch diameter, on a block of wood one foot high. If water is allowed to flow down the tube, so as to leave a sufficient space of air above it, the adhesion between the two ancient elements becomes apparent, directly a little smoke is produced, near the top end of the glass tube resting on the block of wood. The smoke, which has a greater tendency to rise than to fall, is dragged down the glass tube, and accompanies the water as it flows from the higher to the lower level. The same truth is also illustrated in horizontal troughs or tubes through which water is caused to flow. The adhesion between air and glass is so great, that it is absolutely necessary to boil the mercury in the tubes of the best barometers; and if this is not carefully attended to, the adhering air between the glass and mercury gradually ascends to, and destroys, the Torricellian vacuum at the top of the barometer tube. Even after the mercury is boiled, the air will creep up in course of years; and in order to prevent its passage between the glass and quicksilver, it has been recommended, that a platinum ring should be welded on to the end of the glass tube, because mercury has the power of wetting or enfilming the metal platinum, and the two being in close contact, would, as it were, shut the only door by which the air could enter the barometer tube.
- 61. Fig. 81. Model of the apparatus for drawing down air. a, cistern of water, supplied by ball-cock, and kept at one level, so that the water just runs down the sides of the tube, and draws down the air in the centre, b c. The vessel to which the air and water are conveyed by a gutta-percha tube, t. There is another ball- cock to permit the waste water to run away when it reaches a certain level; the end of the pipe always dips some inches into this water, whilst the air escapes from the jet, d.
- 62. CHAPTER VIII. CAPILLARY ATTRACTION. This kind of attraction is termed capillary, in consequence of tubes, of a calibre, or bore, as fine as hair, attracting and retaining fluids. If water is poured into a glass, the surface is not level, but cupped at the edges, where the solid glass exerts its adhesive attraction for the liquid, and draws it from the level. If the glass be reduced to a very narrow tube, having a hair-like bore, the attraction is so great that the water is retained in the tube, contrary to the force of gravitation. Two pieces of flat glass placed close together, and then opened like a book, draw up water between them, on the same principle. A mass of salt put on a plate containing a little water coloured with indigo displays this kind of attraction most perfectly, and the water is quickly drawn up, as shown by the blue colour on the salt. A little solution of the ammonio-sulphate of copper imparts a finer and more distinct blue colour to the salt. A piece of dry Honduras mahogany one inch square, placed in a saucer containing a little turpentine, is soon found to be wet with the oil at the top, which may then be set on fire. Almost every kind of wood possesses capillary tubes, and will float, on account of these minute vessels being filled with air; if, however, the air is withdrawn, then the wood sinks, and by boiling a ball made of beech wood in water, and then placing it under the vacuum of an air pump in other cold water, it becomes so saturated with water that it will no longer float. A remarkable instance of the same kind is mentioned by Scoresby, in which a boat was pulled down by a whale to a great depth in the ocean, and after coming to the surface it was found that the wood would neither swim nor burn, the capillary pores being entirely filled with salt water. A piece of ebony sinks in water on account of its density, closeness, and freedom from air. A gauge made of a piece of oak, with a hole bored in it of one inch diameter, accurately receives a dry plug of willow wood which will not enter the orifice after it is wetted. Millstones are split by inserting wedges of dry hard wood, which are afterwards wetted and swelled, and burst the stone asunder. One of the most curious instances of capillary attraction is shown in the currying of leather, a process which is intended to impart a softness and suppleness to the skin, in order that it may be rendered fit for the manufacture of boots, harness, machine bands, c. The object of the currier is to fill the pores of the leather with oil, and as this cannot be done by merely smearing the surface, he prepares the way for the oil by wetting the leather thoroughly with water, and whilst the skin is damp, oil is rubbed on, and it is then exposed to the air; the water evaporates at ordinary temperatures, but oil does not; the consequence is that the pores of the leather give up the water, which disappears in evaporation, and the oil by capillary attraction is then drawn into the body of the leather, the oil in fact takes the place vacated by the water, and renders the material very supple, and to a considerable extent waterproof. In paper making, the pores of this material, unless filled up or sized, cause the ink to blot or spread by capillary attraction. The porosity of soils is one of the great desideratums of the skilful agriculturist, and drainage is intended to remove the excess of water which would fill the pores of the earth, to the exclusion of the more valuable dews and rains conveying nutritious matter derived from manures and the atmosphere. A cane is an assemblage of small tubes, and if a piece of about six inches in length (cut off, of course, from the joints) be placed in a bottle of turpentine, the oil is drawn up and may be burnt at the top; it is on this principle that indestructible wicks of asbestos, and wire gauze rolled round a centre core, are used in spirit lamps. Oil, wax, and tallow, all rise by capillary attraction in the wicks to the flame, where they are boiled, converted into gas, and burnt.
- 63. Fig. 82. Geber's filter. a. The solution of acetate of lead. b. The dilute sulphuric acid. c. The clear liquid, separated from the sulphate of lead in b. Fig. 83. Prawn syphon. The capillary attraction of skeins of cotton for water was known and appreciated by the old alchemists; and Geber, one of the most ancient of these pioneers of science, and who lived about the seventh century, describes a filter by which the liquid is separated from the solid. This experiment is well displayed by putting a solution of acetate of lead into a glass, which is placed on the highest block of a series of three, arranged as steps. Into this glass is placed the short end of a skein of lamp cotton, previously wetted with distilled water; the long end dips into another glass below, containing dilute sulphuric acid, and as the solution of lead passes into it, a solid white precipitate of sulphate of lead is formed; then another skein of wetted cotton is placed in this glass, the long end of which passes into the last glass, so that the clear liquid is separated and the solid left behind. (Fig. 82.) In this filter the lamp cotton acts as a syphon through the capillary pores which it forms. On the same principle, a prawn may be washed in the most elegant manner (as first shown by the late Duke of Sussex), by placing the tail, after pulling off the fan part, in a tumbler of water, and allowing the head to hang over, when the water is drawn up by capillary attraction, and continues to run through the head. (Fig. 83.) The threads of which linen, cotton, and woollen cloths are made are small cords, and the shrinkage of such textile fabrics, is well known and usually inquired about, when a purchase is made; here again capillary attraction is exerted, and the fabric contracts in the two directions of the warp and woof threads; thus, twenty-seven yards of common Irish linen will permanently shrink to about twenty-six yards in cold water. In these cases the water is attracted into the fibres of the textile material, and causing them to swell, must necessarily shorten their length, just as a dry rope strained between two walls for the purpose of supporting clothes, has been known to draw the hooks after being suddenly wetted and shortened by a shower of rain. In order to tighten a bandage, it is only necessary to wind the dry linen round the limbs as close as possible, and then wet it with water, when the necessary shrinkage takes place. If a piece of dry cotton cloth is tied over one end of a lamp glass, the other may be thrust into, or removed from the basin of water very easily, but when the cotton is wetted, the fibres contract and prevent air from entering, so that the glass retains water just as if it were an ordinary gas jar closed with a glass stopper. A Spanish proverb, expressing contempt, says, go to the well with a sieve, but even this seeming impossibility is surmounted by using a cylinder of wire gauze, which may be filled with water, and by means of the capillary attraction between the meshes of the copper-wire gauze and the water, the whole is retained, and may be carefully lifted from a basin of water; the experiment only succeeds when the air is completely driven out of the interstices of the gauze, and the little cylinder completely filled with water; this may be done by repeatedly sinking and drawing out the cylinder, or still more effectually, by first wetting it with alcohol and then dipping the cylinder in water.
- 64. Fig. 84. a. Basin of water. b. Cylinder of wire gauze closed at both ends with gauze. When full of water it may be lifted from the basin by the handle, c. A balloon, made of cotton cloth, cannot be inflated by means of a pair of bellows, but if the balloon is wetted with water, then it may be swelled out with air just as if it had been made of some air-tight material; hence the principle of varnishing silk or filling the pores with boiled oil, when it is required in the manufacture of balloons. Biscuit ware, porous tubes for voltaic batteries, alcarrazas, or water coolers, are all examples of the same principle. Whilst speaking most favourably of the benevolent labours of many gentlemen (beginning with Mr. Gurney) who have erected Drinking Fountains in London's dusty atmosphere and crowded streets, it must not be forgotten that pious Mohammedans have, in bygone times, already set us the example in this respect; and in the palmy days of many of the Moorish cities, the thirsty citizen could always be refreshed by a draught of cool water from the porous bottles provided and endowed by charitable Mussulmans, and placed in the public streets. Fig 85. Moorish niche and porous earthenware bottle, containing water.
- 65. CHAPTER IX. CRYSTALLIZATION. Fig. 86. Crystals of snow. It has been already stated that the force of cohesion binds the similar particles of substances together, whether they be amorphous or shapeless, crystalline or of a regular symmetrical and mathematical figure. The term crystal was originally applied by the ancients to silica in the form of what is usually termed rock crystal, or Brazilian pebble; and they supposed it to be water which had been solidified by a remarkable intensity of cold, and could not be thawed by any ordinary or summer heat. Indeed, this idea of the ancients has been embodied (to a certain extent) in the shape of artificial ice made by crystallizing large quantities of sulphate of soda, which was made as flat as possible, and upon which skaters were invited to describe the figure of eight, at the usual admittance fee, representing twelve pence. A crystal is now defined to be an inorganic body, which, by the operation of affinity, has assumed the form of a regular solid terminated by a certain number of planes or smooth surfaces. Thousands of minerals are discovered in the crystallized state—such as cubes of iron pyrites (sulphuret of iron) and of fluor spar (fluoride of calcium), whilst numerous saline bodies called salts are sold only in the form of crystals. Of these salts we have excellent examples in Epsom salts (sulphate of magnesia), nitre (nitrate of potash), alum (sulphate of alumina), and potash; the term salt being applied specially to all substances composed of an acid and a base, as also to other combinations of elements which may or may not take a crystalline form. Thus, nitre is composed of nitric acid and potash; the first, even when much diluted, rapidly changes paper, dipped in tincture of litmus and stained blue, to a red colour, whilst potash shows its alkaline nature, by changing paper, stained yellow with tincture of turmeric, to a reddish-brown. The latter paper is restored to its original yellow by dipping it into the dilute nitric acid, whilst the litmus paper regains its delicate blue colour by being passed into the alkaline solution. An acid and an alkali combine and form a neutral salt, such as nitre, which has no action whatever on litmus or turmeric; whilst the element iodine, which is not an acid, unites with the metallic element potassium, and therefore not an alkali, and forms a salt that
- 66. crystallizes in cubes called iodide of potassium. Again, cane sugar, which is composed of charcoal, oxygen, and hydrogen, crystallizes in hard transparent four-sided and irregular six-sided prisms, but is not called a salt. Silica or sand is found crystallized most perfectly in nature in six-sided pyramids, but is not a salt; it is an acid termed silicic-acid. Sand has no acid taste, because it is insoluble in water, but when melted in a crucible with an alkali, such as potash, it forms a salt called silicate of potash. Magnesia, from being insoluble, or nearly so, in water, is all but tasteless, and has barely any alkaline reaction, yet it is a very strong alkaline base; 20.7 parts of it neutralize as much sulphuric acid as 47 of potash. A salt is not always a crystallizable substance, and vice versa. The progress of our chemical knowledge has therefore demanded a wider extension and application of the term salt, and it is not now confined merely to a combination of an acid and an alkali, but is conferred even on compounds consisting only of sulphur and a metal, which are termed sulphur salts. So also in combinations of chlorine, iodine, bromine, and fluorine, with metallic bodies, neither of which are acid or alkaline, the term haloid salts has been applied by Berzelius, from the Greek ( αλς, sea salt, and ειδος form), because they are analogous in constitution to sea salt; and the mention of sea salt again reminds us of the wide signification of the term salt, originally confined to this substance, but now extended into four great orders, as defined by Turner:— Order I. The oxy-salts.—This order includes no salt the acid or base of which is not an oxidised body (ex., nitrate of potash). Order II. The hydro-salts.—This order includes no salt the acid or base of which does not contain hydrogen (ex., chloride of ammonium). Order III. The sulphur salts.—This order includes no salt the electro-positive or negative ingredient of which is not a sulphuret (ex., hydrosulphuret of potassium). Order IV. The haloid salts.—This order includes no salt the electro-positive or negative ingredient of which is not haloidal. (Exs., iodide of potassium and sea salt). To fix the idea of salt still better in the youthful mind, it should be remembered that alabaster, of which works of art are constructed, or marble, or lime-stone, or chalk, are all salts, because they consist of an acid and a base. In order to cause a substance to crystallize it is first necessary to endow the particles with freedom of motion. There are many methods of doing this chemically or by the application of heat, but we cannot by any mechanical process of concentration, compression, or division, persuade a substance to crystallize, unless perhaps we except that remarkable change in wrought or fibrous iron into crystalline or brittle iron, by constant vibration, as in the axles of a carriage, or by attaching a piece of fibrous iron to a tilt hammer. If we powder some alum crystals they will not again assume their crystalline form; if brought in contact there is no freedom of motion. It is like placing some globules of mercury on a plate. They have no power to create motion; their inertia keeps them separated by certain distances, and they do not coalesce; but incline the plate, give them motion, and bring them in contact, they soon unite and form one globule. The particles of alum are not in close contact, and they have no freedom of motion unless they are dissolved in water, when they become invisible; the water by its chemical power destroys the mechanical aggregation of the solid alum far beyond any operation of levigation. The solid alum has become liquid, like water; the particles are now free to move without let or hindrance from friction. A solution, (from the Latin solvo, to loosen) is obtained. The alum must indeed be reduced to minute particles, as they are alike invisible to the eye whether assisted by the microscope or not. No repose will cause the alum to separate; the solvent power of the water opposes gravitation; every part of the solution is equally impregnated with alum, and the particles are diffused at equal distances through the water; the heavy alum is actually drawn up against gravity by the water.
- 67. Fig. 87. r r. Ring-stand. s s. Spirit-lamps. a. Flask containing boiling solution of alum.— Solution. b. Funnel, with a bit of lamp-cotton stuffed in the bottom.—Filtration. c. Evaporating dish.—Evaporation. d. Drop on glass.—Crystallization. Fig. 88. How, then, is the alum to be brought back again to the solid state? The answer is simple enough. By evaporating away the excess of water, either by the application of heat or by long exposure to the atmosphere in a very shallow vessel, the minute atoms of the alum are brought closer together, and crystallization takes place. The assumption of the solid state is indicated by the formation of a thin film (called a pellicle) of crystals, and is further and still more satisfactorily proved by taking out a drop of the solution and placing it on a bit of glass, which rapidly becomes filled with crystals if the evaporation has been carried sufficiently far (Fig. 87). After evaporating away sufficient water, the dish is placed on one side and allowed to cool, when crystals of the utmost regularity of form are produced, and, denoted by a geometrical term, are called octohedral or eight-sided crystals, when in the utmost state of perfection (Fig. 88). The science of crystallography is too elaborate to be discussed at length in a work of this kind; the various terms connected with crystals will therefore only be explained, and experiments given in illustration of the formation of various crystals. When the apices—i.e., the tips or points of crystals—are cut off, they are said to be truncated; and the same change occurs on the edges of numerous crystals. If some of the salt called chloride of calcium in the dry and amorphous state is exposed to the air, it soon absorbs water, or what is termed deliquesces: the same thing occurs with the crystals of carbonate of potash, and if four ounces are weighed out in an evaporating dish, and then exposed for about half an hour to the air, a very perceptible increase in weight is observed by the assistance of the scales and grain weights. Deliquescence is a term from the Latin deliqueo, to melt, and is in fact a gradual melting, caused by the absorption of water from the atmosphere. The reverse of this is illustrated with various crystals, such as Glauber's salt (sulphate of soda), or common washing soda (carbonate of soda); if a fine clear crystal is taken out of the solution, called the mother liquor, in which it has been crystallized, wiped dry, and placed under a glass shade, this salt may remain for a long period without change, but if it receive one scratch from a pin, the door is opened apparently for the escape of the water which it contains, chemically united with the salt, and called water of crystallization; the white crystal gradually swells out, the little quasi sore from the pin-scratch spreads over the whole, which becomes opaque, and crumbling down falls into a shapeless mass of white dust; this change is called efflorescence, from effloresco, to blow as a flower—caused by the abstraction from them of chemically-combined water by the atmosphere. With reference to the preservation of crystals, Professor Griffiths recommends them to be oiled and wiped, and placed under a glass shade, if of a deliquescent nature; or if efflorescent, they are perfectly preserved by placing them under a glass shade with a little water in a cup to keep the air charged with moisture and prevent any drying up of the crystal.
- 68. Fig. 89. Deliquescent crystals may be preserved by placing them, when dry, in naphtha, or any liquor in which they are perfectly insoluble. Some salts, like Glauber's salts, contain so much water of crystallization that when subjected to heat they melt and dissolve in it, and this liquefaction of the solid crystal is called watery fusion. Other salts, such as bay salt, chlorate of potash, c., when heated, fly to pieces, with a sharp crackling noise, which is due sometimes, to the unequal expansion of the crystalline surface, or the sudden conversion of the water (retained in the crystal by capillary attraction) into steam; thus nitre behaves in this manner, and frequently retains water in capillary fissures, although it is an anhydrous salt, or salt perfectly free from combined water. The crackling sound is called decrepitation, and is well illustrated by throwing a handful of bay salt on a clear fire; but this property is destroyed by powdering the crystals. Many substances when melted and slowly cooled concrete into the most perfect crystals; in these cases heat alone, the antagonist to cohesion, is the solvent power. Thus, if bismuth be melted in a crucible, and when cooling, and just as the pellicle (from pellis, a skin or crust) is forming on the surface, if two small holes are instantly made by a rod of iron and the liquid metal poured out from the inside (one of the holes being the entrance for the air, the other the exit for the metal); on carefully breaking the crucible, the bismuth is found to be crystallized in the most lovely cubes. Sulphur, again, may be crystallized in prismatic crystals by pursuing a similar plan; and the great blocks of spermaceti exhibited by wax chandlers in their windows, are crystallized in the interior and prepared on the same principle. There are other modes of conferring the crystalline state upon substances—viz., by elevating them into a state of vapour by the process called sublimation (from sublimis, high or exalted), the lifting up and condensation of the vapour in the upper part of a vessel; a process perfectly distinct from that of distillation, which means to separate drop by drop. Both of these processes are very ancient, and were invented by the Arabian alchemists long antecedent to the seventh century. Examples of sublimation are shown by heating iodine, and especially benzoic acid; with the latter, a very elegant imitation of snow is produced, by receiving the vapour, on some sprigs of holly or other evergreen, or imitation paper snowdrops and crocuses, placed in a tasteful manner under a glass vessel. The benzoic acid should first be sublimed over the sprigs or artificial flowers in a gas jar, which may be removed when the whole is cold, and a clear glass shade substituted for it. (Fig. 89.) All electro deposits on metals are more or less crystalline; and copper or silver may be deposited in a crystalline form by placing a scraped stick of phosphorus in a solution of sulphate of copper or of nitrate of silver. The phosphorus takes away the oxygen from the metal, or deoxidizes the solution, and the copper or silver reappears in the metallic form. The surface of the phosphorus must not be scraped in the air, but under water, when the operation is perfectly safe. A singular and almost instantaneous crystallization can be produced by saturating boiling water with Glauber's salt, of which one ounce and a half of water will usually dissolve about two ounces; having done this, pour the solution, whilst boiling hot, into clean oil flasks, or vials of any kind, previously warmed in the oven, and immediately cork them, or tie strips of wetted bladder, over the orifices of the flasks or vials, or pour into the neck a small quantity of olive oil, or close the neck with a cork through which a thermometer tube has been passed. When cold, no crystallization occurs until atmospheric air is admitted; and it was formerly believed that the pressure of the air effected this object, until some one thought of the oil, and now the theory is modified, and crystallization is supposed to occur in consequence of the water dissolving some air which causes the deposit of a minute crystal, and this being the turning point, the whole becomes solid. However the fact may be explained, it is certain that when the liquid refuses
- 69. a. Gas-jar, with stopper open at first, to be shut when the lamp is withdrawn. b. Wooden stand, with hole to carry the cup c, containing the benzoic acid, heated below by the spirit-lamp, s. f. Flowers or sprigs arranged on pieces of rock or mineral. Fig. 90. a. The inner cylinder which contains the freezing mixture. b b. The outer one containing spring water. c c. The ice slipping away from the inner cylinder. to crystallize on the admission of air, the solidification occurs directly a minute crystal of sulphate of soda, or Glauber's salt, is dropped into the vessel. When the crystallization is accomplished, the whole mass is usually so completely solidified, that on inverting the vessel, not a drop of liquid falls out. It may be observed that the same mass of salt will answer any number of times the same purpose. All that is necessary to be done, is to place the vial or flask, in a saucepan of warm water, and gradually raise it to the boiling point till the salt is completely liquefied, when the vessel must be corked and secured from the air as before. When the solidification is produced much heat is generated, which is rendered apparent by means of a thermometer, or by the insertion of a copper wire into the pasty mass of crystal in the flask, and then touching an extremely thin shaving or cutting of phosphorus, dried and placed on cotton wool. Solidification in all cases produces heat. Liquefaction produces cold. In Masters's freezing apparatus certain measured quantities of crystallized sal-ammoniac, nitre, and nitrate of ammonia, are placed in a metallic cylinder, surrounded with a small quantity of spring water contained in an outer vessel. Directly the crystals are liquefied by the addition of water, intense cold is produced, which freezes the water and forms an exact cast of the inner cylinder in ice, and this may afterwards be removed, by pouring away the liquefied salts, and filling the inner cylinder, with water of the same temperature as the air, which rapidly thaws the surrounding ice, and allows it to slip off into any convenient vessel ready to receive it. (Fig. 90.) For an ingenious method of obtaining large and perfect crystals of almost any size, experimentalists are indebted to Le Blanc. His method consists in first procuring small and perfect crystals—say, octohedra of alum—and then placing them in a broad flat- bottomed pan, he pours over the crystals a quantity of saturated solution of alum, obtained by evaporating a solution of alum until a drop taken out crystallizes on cooling. The positions of the crystals are altered at least once a day with a glass rod, so that all the faces may be alternately exposed to the action of the solution, for the side on which the crystal rests, or is in contact with the vessel, never receives any increment. The crystals will thus gradually grow or increase in size, and when they have done so for some time, the best and most symmetrical, may be removed and placed separately, in vessels containing some of the same saturated solution of alum, and being constantly turned they may be obtained of almost any size desired. Unless the crystals are removed to fresh solutions, a reaction takes place, in consequence of the exhaustion of the alum from the water, and the crystal is attacked and dissolved. This action is first perceptible on the edges and angles of the crystal; they become blunted and gradually lose their shape altogether. By this method crystals may be made to grow in length or breadth—the former when they are placed upon their sides, the latter if they be made to stand upon their bases. On Le Blanc's principle, beautiful crystal baskets are made with alum, sulphate of copper, and bichromate of potash. The baskets are usually made of covered copper wire, and when the salts
- 70. crystallize on them as a nucleus or centre, they are constantly removed to fresh solutions, so that the whole is completely covered, and red, white, and blue sparkling crystal baskets formed. They will retain their brilliancy for any time, by placing them under a glass shade, with a cup containing a little water. The sketch below affords an excellent illustration of some of Nature's remarkable concretions in the peculiar columnar structure of basalt. Fig. 91. The Giant's Causeway.
- 71. CHAPTER X. CHEMISTRY. Fig 92 Alchemists at work There is hardly any kind of knowledge which has been so slowly acquired as that of chemistry, and perhaps no other science has offered such fascinating rewards to the labour of its votaries as the philosopher's stone, which was to produce an unfailing supply of gold; or the elixir of life, that was to give the discoverer of the gold-making art the time, the prolonged life, in which he might spend and enjoy it. Hundreds of years ago Egypt was the great depository of all learning, art, and science, and it was to this ancient country that the most celebrated sages of antiquity travelled. Hermes, or Mercurius Trismegistus, the favourite minister of the Egyptian king Osiris, has been celebrated as the inventor of the art of alchemy, and the first treatise upon it has been attributed to Zosymus, of Chemnis or Panopolis. The Moors who conquered Spain were remarkable for their learning, and the taste and elegance with which they designed and carried out a new style of architecture, with its lovely Arabesque ornamentation. They were likewise great followers of the art of alchemy, when they ceased to be conquerors, and became more reconciled to the arts of peace. Strange that such a people, thirsting as they did in after years for all kinds of knowledge, should have
- 72. destroyed, in the persons of their ancestors, the most numerous collection of books that the world had ever seen: the magnificent library of Alexandria, collected by the Ptolemies with great diligence and at an enormous expense, was burned by the orders of Caliph Omar; whilst it is stated that the alchemical works had been previously destroyed by Diocletian in the fourth century, lest the Egyptians should acquire by such means sufficient wealth to withstand the Roman power, for gold was then, as it is now, the sinews of war. Eastern historians relate the trouble and expense incurred by the succeeding Caliphs, who, resigning the Saracenic barbarism of their ancestors, were glad to collect from all parts the books which were to furnish forth a princely library at Bagdad. How the learned scholar sighs when he reads of seven hundred thousand books being consigned to the ignominious office of heating forty thousand baths in the capital of Egypt, and of the magnificent Alexandrian Library, a mental fuel for the lamp of learning in all ages, consumed in bath furnaces, and affording six months' fuel for that purpose. The Arabians, however, made amends for these barbarous deeds in succeeding centuries, and when all Europe was laid waste under the iron rule of the Goths, they became the protectors of philosophy and the promoters of its pursuits; and thus we come to the seventh century, in which Geber, an Arabian prince lived, and is stated to be the earliest of the true alchemists whose name has reached posterity. Without attempting to fill up the alchemical history of the intervening centuries, we leap forward six hundred years, and now find ourselves in imagination in England, with the learned friar, Roger Bacon, a native of Somersetshire, who lived about the middle of the thirteenth century; and although the continual study of alchemy had not yet produced the stone, it bore fruit in other discoveries, and Roger Bacon is said, with great appearance of truth, to have discovered gunpowder, for he says in one of his works:—From saltpetre and other ingredients we are able to form a fire which will burn to any distance; and again alluding to its effects, a small portion of matter, about the size of the thumb, properly disposed, will make a tremendous sound and coruscation, by which cities and armies might be destroyed. The exaggerated style seems to have been a favourite one with all philosophers, from the time of Roger Bacon to that of Muschenbroek of the University of Leyden, who accidentally discovered the Leyden jar in the year 1746, and receiving the first shock, from a vial containing a little water, into which a cork and nail had been fitted, states that he felt himself struck in his arms, shoulders, and breast, so that he lost his breath, and was two days before he recovered from the effects of the blow and the terror; adding, that he would not take a second shock for the kingdom of France. Disregarding the numerous alchemical events occurring from the time of Roger Bacon, we again advance four hundred years—viz., to the year 1662, when, on the 15th of July, King Charles II. granted a royal charter to the Philosophical Society of Oxford, who had removed to London, under the name of the Royal Society of London for Promoting Natural Knowledge, and in the year 1665 was published the first number of the Philosophical Transactions; this work contains the successive discoveries of Mayow, Hales, Black, Leslie, Cavendish, Lavoisier, Priestley, Davy, Faraday; and since the year 1762 has been regularly published at the rate of one volume per annum. With this preface proceed we now to discuss some of the varied phenomena of chemical attraction, or what is more correctly termed CHEMICAL AFFINITY. The above title refers to an endless series of changes brought about by chemical combinations, all of which can be reduced to certain fixed laws, and admit of a simple classification and arrangement. A mechanical aggregation, however well arranged, can be always distinguished from a chemical one. Thus, a grain of gunpowder consists of nitre, which can be washed away with boiling water, of sulphur, which can be sublimed and made to pass away as vapour, of charcoal, which remains behind after the previous processes are complete; this mixture has been perfected by a careful proportion of the respective ingredients, it has been wetted, and ground, and pressed, granulated, and finally dried; all these mechanical processes have been so well carried out that each grain, if analysed, would be similar to the other; and yet it is, after all, only a mechanical aggregation, because the sulphur, the
- 73. charcoal, and the nitre are unchanged. A grain of gunpowder moistened, crushed, and examined by a high microscopic power, would indicate the yellow particles of sulphur, the black parts of charcoal, whilst the water filtered from the grain of powder and dried, would show the nitre by the form of the crystal. On the other hand, if some nitre is fused at a dull red heat in a little crucible, and two or three grains of sulphur are added, they are rapidly oxidized, and combine with the potash, forming sulphate of potash; and after this change a few grains of charcoal may be added in a similar manner, when they burn brightly, and are oxidized and converted into carbonic acid, which also unites in like manner with the potash, forming carbonate of potash; so that when the fused nitre is cooled and a few particles examined by the microscope, the charcoal and sulphur are no longer distinguishable, they have undergone a chemical combination with portions of the nitre, and have produced two new salts, perfectly different in taste, gravity, and appearance from the original substances employed to produce them. Hence chemical combination is defined to be that property which is possessed by one or more substances, of uniting together and producing a third or other body perfectly different in its nature from either of the two or more generating the new compound. To return to our first experiment with the gunpowder: take sulphur, place some in an iron ladle, heat it over a gas flame till it catches fire, then ascend a ladder, and pour it gently, from the greatest height you can reach, into a pail of warm water: if this experiment is performed in a darkened room a magnificent and continuous stream of fire is obtained, of a blue colour, without a single break in its whole length, provided the ladle is gradually inclined and emptied. The substance that drops into the warm water is no longer yellow and hard, but is red, soft, and plastic; it is still sulphur, though it has taken a new form, because that element is dimorphous (δις twice, and μορφη a form), and, Proteus- like, can assume two forms. Take another ladle, and melt some nitre in it at a dull red heat, then add a small quantity of sulphur, which will burn as before; and now, after waiting a few minutes, repeat the same experiment by pouring the liquid from the steps through the air into water; observe it no longer flames, and the substance received into the water is not red and soft and plastic, but is white, or nearly so, and rapidly dissolves away in the water. The sulphur has united with the oxygen of the nitre and formed sulphuric acid, which combines with the potash and forms sulphate of potash; here, then, oxygen, sulphur, and potassium, have united and formed a salt in which the separate properties of the three bodies have completely disappeared; to prove this, it is only necessary to dissolve the sulphate of potash in water, and after filtering the solution, or allowing it to settle, till it becomes quite clear and bright, some solution of baryta may now be added, when a white precipitate is thrown down, consisting of sulphate of baryta, which is insoluble in nitric or other strong acids. The behaviour of a solution of sulphate of potash with a nitrate of baryta may now be contrasted with that of the elements it contains; on the addition of sulphur to a solution of nitrate of baryta no change whatever takes place, because the sulphur is perfectly insoluble. If a stream of oxygen gas is passed from a bladder and jet through the same test, no effect is produced; the nitrate of baryta has already acquired its full proportion of oxygen, and no further addition has any power to change its nature; finally, if a bit of the metal potassium is placed in the solution of nitrate of baryta it does not sink, being lighter than water, and it takes fire; but this is not in any way connected with the presence of the test, as the same thing will happen if another bit of the metal is placed in water—it is the oxygen of the latter which unites rapidly with the potassium, and causes it to become so hot that the hydrogen, escaping around the little red-hot globules, takes fire; moreover, the fact of the combustion of the potassium under such circumstances is another striking proof of the opposite qualities of the three elements—sulphur, oxygen, and potassium—as compared with the three chemically combined and forming sulphate of potash. The same kind of experiment may be repeated with charcoal; if some powdered charcoal is made red-hot, and then puffed into the air with a blowing machine, numbers of sparks are produced, and the charcoal burns away and forms carbonic acid gas, a little ash being left behind; but if some more nitre be heated in a ladle, and charcoal added, a brilliant deflagration (deflagro, to burn) occurs, and the charcoal, instead of passing away in the air as carbonic acid, is now retained in the same shape, but firmly and chemically united with the potash of the nitre, forming carbonate of potash, or pearl-ash, which is not black and insoluble in water and acids like charcoal, but is white, and not only soluble in water, but is most rapidly attacked by acids with
- 74. effervescence, and the carbon escapes in the form of carbonic acid gas. Thus we have traced out the distinction between mechanical aggregation and chemical affinity, taking for an example the difference between gunpowder as a whole (in which the ingredients are so nicely balanced that it is almost a chemical combination), and its constituents, sulphur, charcoal, and nitre, when they are chemically combined; or, in briefer language, we have noticed the difference between the mechanical mixture, and some of the chemical combinations, of three important elements. Our very slight and partial examination of three simple bodies does not, however, afford us any deep insight into the principles of chemistry; we have, as it were, only mastered the signification of a few words in a language; we might know that chien was the French for dog, or cheval horse, or homme man; but that knowledge would not be the acquisition of the French language, because we must first know the alphabet, and then the combination of these letters into words; we must also acquire a knowledge of the proper arrangement of these words into sentences, or grammar, both syntax and prosody, before we can claim to be a French scholar: so it is with chemistry—any number of isolated experiments with various chemical substances would be comparatively useless, and therefore the alphabet of chemistry, or table of simple elements, must first be acquired. These bodies are understood to be solids, fluids, and gases, which have hitherto defied the most elaborate means employed to reduce them into more than one kind of matter. Even pure light is separable into seven parts—viz., red, orange, yellow, green, blue, indigo, and violet; but the elements we shall now enumerate are not of a compound, but, so far as we know, of an absolutely simple or single nature; they represent the boundaries, not the finality, of the knowledge that may be acquired respecting them. The elements are sixty-four in number, of which about forty are tolerably plentiful, and therefore common; whilst the remainder, twenty-four, are rare, and for that reason of a lesser utility: whenever Nature employs an element on a grand scale it may certainly be called common, but it generally works for the common good of all, and fulfils the most important offices. CLASSIFICATION OF THE ALPHABET OF CHEMISTRY. 13 Non-Metallic Bodies. Name. Symbol. Combining proportion or atomic weight. 1.Oxygen O = 8 2.Hydrogen H = 1 3.Nitrogen N = 14 4.Chlorine Cl = 35.5 5.Iodine I = 127.1 6.Bromine Br = 80.0 7.Fluorine F = 18.9 8.Carbon C = 6 9.Boron B = 10.9 10.Sulphur Sv = 16 11.Phosphorus P = 32 12.Silicon Si = 21.3 13.Selenium Se = 39.5 51 Metals. 1. Aluminium Al = 13.7 2. Antimony Sb = 129 3. Arsenic As = 75
- 75. 4. Barium Ba = 68.5 5. Bismuth Bi = 213 6. Cadmium Cd = 56 7. Calcium Ca = 20 8. Cerium Ce = 47 9. Chromium Cr = 26.7 10. Cobalt Co = 29.5 11. Copper Cu = 31.7 12. Donarium 13. Didymium D 14. Erbium E 15. Gold Au = 197 16. Glucinum Gl 17. Iron Fe = 28 18. Ilmenium Il 19. Iridium Ir = 99 20. Lead Pb = 103.7 21. Lanthanium La 22. Lithium Li = 6.5 23. Magnesium Mg = 12.2 24. Manganese Mn = 27.6 25. Mercury Hg = 100 26. Molybdenum Mo = 46 27. Nickel Ni = 29.6 28. Norium 29. Niobium Nb 30. Osmium Os = 99.6 31. Platinum Pt = 98.7 32. Potassium K = 39.2 33. Palladium Pd = 53.3 34. Pelopium Pe 35. Rhodium R = 52.2 36. Rhuthenium Ru = 52.2 37. Silver Ag = 108.1 38. Sodium Na = 23 39. Strontium Sr = 43.8 40. Tin Sn = 59 41. Tantalum Ta = 184 42. Tellurium Te = 64.2 43. Terbium Tb 44. Thorium Th = 59.6 45. Titanium Ti = 25 46. Tungsten W[A]= 95 47. Uranium U = 60 48. Vanadium V = 68.6
- 76. 49. Yttrium Y 50. Zinc Zn = 32.6 51. Zirconium Zr = 22.4 (N.B. The elements printed in italics are at present unimportant.) [A] From the mineral Wolfram, and now exceedingly valuable, as when alloyed with iron it is harder than, and will bore through steel. A few words will suffice to explain the meaning of the terms which head the names, letters, and numbers of the Table of Elements. The names of the elements have very interesting derivations, which it is not the object of this work to go into; the symbols are abbreviations, ciphers of the simplest kind, to save time and trouble in the frequent repetition of long words, just as the signs + plus, and - minus, are used in algebraic formulæ. For instance—the constant recurrence of water in chemical combinations must be named, and would involve the most tedious repetition; water consists of oxygen and hydrogen, and by taking the first letter of each word we have an instructive symbol, which not only gives us an abbreviated term for water, but also imparts at once a knowledge of its composition by the use of the letters, HO. Again, to take a more complex example, such as would occur in the study of organic chemistry—a sentence such as the hydrated oxide of acetule, is written at once by C4H4O2, the figures referring to the number of equivalents of each element—viz., 4 equivalents of C, the symbol for carbon, 4 of H (hydrogen), and 2 of O (oxygen). The long word paranaphthaline, a substance contained in coal tar, is disposed of at once with the symbols and figures C30H12. The figures in the third column are, however, the most interesting to the precise and mathematically exact chemist. They represent the united labours of the most painstaking and learned chemists, and are the exact quantities in which the various elements unite. To quote one example: if 8 parts by weight of oxygen—viz., the combining proportions of that element—are united with 1 part by weight of hydrogen, also its combining number, the result will be 9 parts by weight of water; but if 8 parts of oxygen and 2 parts of hydrogen were used, one only of the latter could unite with the former, and the result would be the formation again of 9 parts of water, with an overplus of 1 equivalent of hydrogen. It is useless to multiply examples, and it is sufficient to know that with this table of numbers the figures of analysis are obtained. Supposing a substance contained 27 parts of water, and the oxygen in this had to be determined, the rule of proportion would give it at once, 9: 27:: 8: 24. 9 parts of water are to 27 parts as 8 of oxygen (the quantity contained in 9 parts of water) are to the answer required—viz., 24 of oxygen. The names, symbols, and combining proportions being understood, we may now proceed with the performance of many interesting CHEMICAL EXPERIMENTS. As the permanent gases head the list, they will first engage our attention, beginning with the element oxygen—Symbol O, combining proportion 8. There is nothing can give a better idea of the enormous quantity of oxygen present in the animal, vegetable, and mineral kingdoms, than the statement that it represents one-third of the weight of the whole crust of the globe. Silica, or flint, contains about half its weight of oxygen; lime contains forty per cent.; alumina about thirty-three per cent. In these substances the element oxygen remains inactive and powerless, chained by the strong fetters of chemical affinity to the silicium of the flint, the calcium of the lime, and the aluminum of the alumina. If these substances are heated by themselves they will not yield up the large quantity of oxygen they contain.
- 77. Nature, however, is prodigal in her creation, and hence we have but to pursue our search diligently to find a substance or mineral containing an abundance of oxygen, and part of which it will relinquish by what used to be called by the old alchemists the torture of heat. Such a mineral is the black oxide of manganese, or more correctly the binoxide of manganese, which consists of one combining proportion of the metal manganese—viz., 27.6, and two of oxygen—viz., 8 × 2 = 16. If three proportions of the binoxide of manganese are heated to redness in an iron retort, they yield one proportion (equal to 8) of oxygen, and all that has just been explained by so many words is comprehended in the symbols and figures below:— 3 MnO2 = Mn3O4 + O. Thus the 3 MnO2 represent the three proportions of the binoxide of manganese before heat is applied, whilst the sign =, the sign of equation (equal to), is intended to show that the elements or compounds placed before it produce those which follow it; hence the sequel Mn3O4 + O shows that another compound of the metal and oxygen is produced, whilst the + O indicates the liberated oxygen gas. The iron retort employed to hold the mineral should be made of cast iron in preference to wrought iron, as the latter is very soon worn out by contact with oxygen at a red heat. A gun-barrel will answer the purpose for an experiment on the small scale, to which must be adapted a cock and piece of pewter tubing. Such a make-shift arrangement may do very well when nothing better offers; but as a question of expense, it is probably cheaper in the end to order of Messrs. Simpson and Maule, or of Messrs. Griffin, or of Messrs. Bolton, a cast-iron bottle, or cast-iron retort, as it is termed, of a size sufficient to prepare two gallons of oxygen from the binoxide of manganese, which, with four feet of iron conducting-pipe, and connected to the bottle with a screw, does not cost more than six shillings—an enormous dip, perhaps, in the juvenile pocket, and therefore we shall indicate presently a still cheaper apparatus for the same purpose. (Fig. 93.) Fig. 93. a. The iron bottle, containing the black oxide of manganese, with pipe passing to the pneumatic trough, b b, in which is fixed a shelf, c, perforated with a hole, under which the end of the pipe is adjusted, and the gas passes into the gas-jar, d. The oxygen is conveyed to a square tin box provided with a shelf at one end, perforated with several holes at least one inch in diameter, called the pneumatic trough; any wooden trough, butter or wash- tub, foot-pan or bath, provided with a shelf, may be raised by the same title to the dignity of a piece of chemical apparatus. The gas jar must be filled with water by withdrawing the stopper and pressing it down into the trough, and when the neck is below the level of the water, the stopper is again inserted, and the jar with the water therein contained lifted steadily on to the shelf, the entry of atmospheric air being prevented by keeping the lower part of the gas jar, called the welt, under the water. Sometimes the pneumatic trough contains so small a quantity of water that on raising the gas jar to the shelf the liquid does not cover the bottom, and the air rushes up in large bubbles. Under these circumstances it is better to provide a gallon stone jug full of water, so that when the jar is
- 78. being raised to the shelf it may be thrust into the trough (on the same principle as the crow and the pitcher in the fable), and thus by its bulk (as the stones in the pitcher) raise the water to the proper level. When the gas jar is about half filled with gas the jug may be withdrawn. This arrangement saves the trouble of constantly adding and baling out water from the pneumatic trough. (Fig. 94.) Fig. 94. a a. Pneumatic trough, with gas jar raised to shelf; bubbles of air are rushing in at b, as the level of the water is below the shelf—viz., at c c. d d. Same trough and gas jar with water kept over the shelf by the introduction of the stone pitcher e, full of water. There are other solid oxygenized bodies in which the affinities are less powerful, and hence a lower degree of heat suffices to liberate the oxygen gas, and one of the most useful in this respect is the salt termed chlorate of potash. If the substance is heated by itself, the temperature required to expel the oxygen is almost as high as that demanded for the black oxide of manganese; but, strange to say, if the two substances are reduced to powder, and mixed in equal quantities by weight, then a very moderate increase of heat is sufficient to cause the chlorate of potash to give up its oxygen, whilst the oxide of manganese undergoes no change whatever. It seems to fulfil only a mechanical office— possibly that of separating each particle of chlorate of potash from the other, so that the heat attacks the substance in detail, just as a solid square of infantry might repel almost any attack, whilst the same body dispersed over a large space might be of little use; so with the chlorate of potash, which undergoes rapid decomposition when mixed with and divided amongst the particles of the oxide of manganese; less so with the red oxide of iron, and still less with sand or brick-dust. (Fig. 95.) Fig. 95.
- 79. Preparation of oxygen from chlorate of potash and oxide of manganese. KO.ClO5 = KCl + O6. This curious fact is explained usually by reference to what is called catalytic action, or decomposition by contact (κατα, downwards, and λυω, I unloosen), being a power possessed by a body of resolving another into a new compound without undergoing any change itself. To make this term still clearer, we may notice another example in linen rags, which may be exposed for any length of time to the action of water without fear of conversion into sugar; if, however, oil of vitriol is first added to the linen rags, and they are subsequently digested at a proper temperature with water, then the rags are converted into sugar (the author has seen a specimen made of an old shirt); but, curious to relate, the oil of vitriol is unchanged in the process, and if the process be commenced with a pound of acid, the same quantity is discoverable at the end of the chemical decomposition of the linen rags, and their conversion into sugar. If a mixture of equal parts of oxide of manganese and chlorate of potash is placed in a clean Florence flask, with a cork, and pewter, or glass tube attached, great quantities of oxygen are quickly liberated, on the application of the heat of a spirit lamp. Such a retort would cost about fourpence, and if the flask is broken in the operation it can be easily replaced by another, value one penny, as the same cork and tube will generally suit a number of these cheap glass vessels. Corks may always be softened by using either a proper cork squeezer, or by placing them under a piece of board or a flat surface, and rolling and pressing the cork till quite elastic. Whilst fitting the latter into the neck of a flask, it is perhaps safer to hold the thin and fragile vessel in a cloth, so that if the flask breaks the chemical experiment may not be arrested for many days by the severe cutting and wounding of the fingers. After the cork is fitted, it is to be removed from the flask and bored with a cork borer. This useful tool is sold in complete sets to suit all sizes of glass tubes, and the pewter or glass being inserted, the flask and tube will be ready for use, provided the tube is bent to the proper curve. This is easy enough to perform with the pewter, but not quite so easy with the glass tube, which must be held over the flame of a spirit lamp till soft, and then bent very gradually to the proper curve. If a short length of the glass tube is heated, it bends too sharply, and the convexity of the glass is flattened, whilst the internal diameter of the tube is lessened, so that at least three inches in length should be warmed, and the heat must not be continued in one place only, but should be maintained in the direction of the bend, the whole manipulation being conducted without any hurry. (Fig. 96.) Fig. 96. a. The cork squeezer. b. The cork borers. c. The operation of bending the glass tube over the flame of the spirit-lamp. d. The neck of the flask, with cork and tube bent and fitted complete for use.
- 80. Fig. 97. Having filled a gas jar with oxygen, it may be removed from the pneumatic trough by sliding it into a plate under the surface of the water, and to prevent the stopper being thrust out accidentally from the jar by the upward pressure of the gas, whilst a little compressed, during the act of passing it into the plate, it is advisable to hold the stopper of the jar firmly but gently, so that it cannot slip out of its place. A number of jars of oxygen may be prepared and arranged in plates, all of which of course must contain a little water, and enough to cover the welt of the jar. EXPERIMENTS WITH OXYGEN GAS. This gas was originally discovered by Priestley, in August, 1774, and was first obtained by heating red precipitate—i.e., the red oxide of mercury. HgO = Hg + O. We leave these symbols and figures to be deciphered by the youthful philosopher with the aid of the table of elements, c., and return to the experiments. There are certain thin wax tapers like waxed cord, called bougies, which can be bent to any shape, and are very convenient for experiments with the gases. If one of these tapers is bent as in Fig. 97, then lighted and allowed to burn for some minutes, a long snuff is gradually formed, which remains in a state of ignition when the flame of the taper is blown out. On plunging this into a jar of oxygen, it instantly re-lights with a sort of report, and burns with greatly-increased brilliancy, as described by Dr. Priestley in his first experiment with this gas, and so elegantly repeated by Professor Brande in his refined dissertation on the progress of chemical science. The 1st of August, 1774, is a red-letter day in the annals of chemical philosophy, for it was then that Dr. Priestley discovered dephlogisticated air. Some, sporting in the sunshine of rhetoric, have called this the birthday of pneumatic chemistry; but it was even a more marked and memorable period; it was then (to pursue the metaphor) that this branch of science, having eked out a sickly and infirm infancy in the ill-managed nursery of the early chemists, began to display symptoms of an improving constitution, and to exhibit the most hopeful and unexpected marks of future importance. The first experiment, which led to a very satisfactory result, was concluded as follows:—A glass jar was filled with quicksilver, and inserted in a basin of the same; some red precipitate of quicksilver was then introduced, and floated upon the quicksilver in the jar; heat was applied to it in this situation with a burning-lens, and to use Priestley's own words, I presently found that air was expelled from it very readily. Having got about three or four times as much as the bulk of my materials, I admitted water into it, and found that it was not imbibed by it. But what surprised me more than I can well express was, that a candle burned in this air with a remarkably vigorous flame, very much like that enlarged flame with which a candle burns in nitrous air exposed to iron or lime of sulphur (i.e., laughing gas); but as I had got nothing like this remarkable appearance from any kind of air besides this peculiar modification of nitrous air, and I knew no nitrous acid was used in the preparation of mercurius calcinatus, I was utterly at a loss how to account for it. (Fig. 98.)
- 81. Fig. 98. a. Glass vessel full of mercury, containing the red precipitate at the top, and standing in the dish b, also containing mercury. c. The burning-glass concentrating the sun's rays on the red precipitate, being Priestley's original experiment. Second Experiment. The term oxygen is derived from the Greek (οζυς, acid, and γενναω, I give rise to), and was originally given to this element by Lavoisier, who also claimed its discovery; and if this honour is denied him, surely he has deserved equal scientific glory in his masterly experiments, through which he discovered that the mixture of forty-two parts by measure of azote, with eight parts by measure of oxygen, produced a compound precisely resembling our atmosphere. The name given to oxygen was founded on a series of experiments, one of which will now be mentioned. Place some sulphur in a little copper ladle attached to a wire, and called a deflagrating spoon, passed through a round piece of zinc or brass plate and cork, so that the latter acts as an adjusting arrangement to fix the wire at any point required. The combustion of the sulphur, previously feeble, now assumes a remarkable intensity, and a peculiar coloured light is generated, whilst the sulphur unites with the oxygen, and forms sulphurous acid gas. It produces, in fact, the same gas which is formed by burning an ordinary sulphur match. This compound is valuable as a disinfectant, and is a very important bleaching agent, being most extensively employed in the whitening of straw employed in the manufacture of straw bonnets. It is an acid gas, as Lavoisier found, and this property may be detected by pouring a little tincture of litmus into the bottom of the plate in which the gas jar stands. The blue colour of the litmus is rapidly changed to red, and it might be thought that no further argument could possibly be required to prove that oxygen was the acidifying agent, the more particularly as the result is the same in the next illustration.
- 82. Fig. 99. a. The deflagrating spoon, b. The cork. c. The zinc, or brass, or tin plate. d d. The gas jar. Third Experiment. Cut a small piece from an ordinary stick of phosphorus under water, take care to dry it properly with a cloth, and after placing it in a deflagrating spoon, remove the stopper from the gas-jar, as there is no fear of the oxygen rushing away, because it is somewhat heavier than atmospheric air; and then, after placing the spoon with the phosphorus in the neck of the jar, apply a heated wire and pass the spoon at once into the middle of the oxygen; in a few seconds a most brilliant light is obtained, and the jar is filled with a white smoke; as this subsides, being phosphoric acid, and perfectly soluble in water, the same litmus test may be applied, when it is in like manner changed to red. The acid obtained is one of the most important constituents of bone. Fourth Experiment. A bit of bark-charcoal bound round with wire is set on fire either by holding it in the flame of a spirit- lamp, or by attaching a small piece of waxed cotton to the lower part, and igniting this; the charcoal may then be inserted into a bottle of oxygen, when the most brilliant scintillations occur. After the combustion has ceased and the whole is cool, a little tincture of litmus may also be poured in and shaken about, when it likewise turns red, proving for the third time the generation of an acid body, called carbonic acid—an acid, like the others already mentioned, of great value, and one which Nature employs on a stupendous scale as a means of providing plants, c., with solid charcoal. Carbonic acid, a virulent poison to animal life, is, when properly diluted, and as contained in atmospheric air, one of the chief alimentary bodies required by growing and healthy plants. In three experiments acid bodies have been obtained; can we speculate on the result of the next?
- 83. Fifth Experiment. Into a deflagrating spoon place a bit of potassium, set this on fire by holding it in the spoon in the flame of a spirit-lamp, and then rapidly plunge the burning metal into a bottle of oxygen. A brilliant ignition occurs in the deflagrating spoon for a few seconds, and there is little or no smoke in the jar. The product this time is a solid, called potash, and if this be dissolved in water and filtered, it is found to be clear and bright, and now on the addition of a little tincture of litmus to one half of the solution, it is wholly unaffected, and remains blue; but if with the other half a small quantity of tincture of turmeric is mixed, it immediately changes from a bright yellow solution to a reddish-brown, because turmeric is one of the tests for an alkali; and thus is ascertained by the help of this and other tests that the result of the combustion is not an acid, but an alkali. The experiment is made still more satisfactory by burning another bit of potassium in oxygen and dissolving the product in water, and if any portion of the reddened liquid derived from the sulphurous, phosphoric, and carbonic acids taken from the previous experiments, be added to separate portions of the alkaline solution, they are all restored to their original blue colour, because an acid is neutralized by an alkali; and the experiment is made quite conclusive by the restoration of the reddened turmeric to a bright yellow on the addition of a solution of either of the three acids already named. Moreover, an acid need not contain a fraction of oxygen, as there is a numerous class of hydracids, in which the acidifying principle is hydrogen instead of oxygen, such as the hydrochloric, hydriodic, hydro-bromic, and hydrofluoric acids. Sixth Experiment. A piece of watch-spring is softened at one end, by holding it in the flame of a spirit-lamp, and allowing it to cool. A bit of waxed cotton is then bound round the softened end, and after being set on fire, is plunged into a gas jar containing oxygen; the cotton first burns away, and then the heat communicates to the steel, which gradually takes fire, and being once well ignited, continues to burn with amazing rapidity, forming drops of liquid dross, which fall to the bottom of the plate—and also a reddish smoke, which condenses on the sides of the jar; neither the dross which has dropped into the plate, nor the reddish matter condensed on the jar, will affect either tincture of litmus or turmeric; they are neither acid nor alkaline, but neutral compounds of iron, called the sesquioxide of iron (Fe2O3), and the magnetic oxide (Fe3O4 = FeO.Fe2O3). Seventh Experiment. Some oxygen gas contained in a bladder provided with a proper jet may be squeezed out, and upon, some liquid phosphorus contained in a cup at the bottom of a finger glass full of boiling water, when a most brilliant combustion occurs, proving that so long as the principle is complied with—viz., that of furnishing oxygen to a combustible substance—it will burn under water, provided it is insoluble, and possesses the remarkable affinity for oxygen which belongs to phosphorus. The experiment should be performed with boiling water, to keep the phosphorus in the liquid state; and it is quite as well to hold a square foot of wire gauze over the finger glass whilst the experiment is being performed. (Fig. 100.) Fig. 100. a. Bladder containing oxygen, provided with a stop-cock and jet leading to, b, b. Finger glass containing boiling water. c. The cup of melted phosphorus under the water. The gas escapes from the bladder
- 84. when pressed. Eighth Experiment. Oxygen is available from many substances when they are mixed with combustible substances, and hence the brilliant effects produced by burning a mixture of nitre, meal powder, sulphur, and iron or steel filings; the metal burns with great brilliancy, and is projected from the case in most beautiful sparks, which are long and needle-shaped with steel, and in the form of miniature rosettes with iron filings; it is the oxygen from the nitre that causes the combustion of the metal, the other ingredients only accelerate the heat and rate of ignition of the brilliant iron, which is usually termed a gerb. Ninth Experiment. A mixture of nitrate of potash, powdered charcoal, sulphur, and nitrate of strontium, driven into a strong paper case about two inches long, and well closed at the end with varnish, being quite waterproof, may be set on fire, and will continue to burn under water until the whole is consumed; the only precaution necessary being to burn the composition from the case with the mouth downward, and if the experiment is tried in a deep glass jar it has a very pleasing effect. (Fig. 101.) Fig. 101. a. Case of red fire burning downwards, and attached by a copper wire to a bit of leaden pipe b, to sink it. c c. Jar containing water. The red-fire composition is made by mixing nitrate of strontia 40 parts by weight, flowers of sulphur 13 parts, chlorate of potash 5 parts, sulphuret of antimony 4 parts. These ingredients must first be well powdered separately, and then mixed carefully on a sheet of paper with a paper-knife. They are liable to explode if rubbed together in a mortar, on account of the presence of sulphur and chlorate of potash, and the composition, if kept for any time, is liable to take fire spontaneously. Tenth Experiment. Some zinc is melted in an iron ladle, and made quite red hot; if a little dry nitre is thrown upon the surface, and gently stirred into the metal, it takes fire with the production of an intense white light, whilst large quantities of white flakes ascend, and again descend when cold, being the oxide of zinc,
- 85. and called by the alchemists the Philosopher's Wool (ZnO). In this experiment the oxygen from the nitre effects the oxidation of the metal zinc. Eleventh Experiment. A mixture of four pounds of nitre with two of sulphur and one and a half of lamp black produces a most beautiful and curious fire, continually projected into the air as sparks having the shape of the rowel of a spur, and one that may be burnt with perfect safety in a room, as the sparks consume away so rapidly, in consequence of the finely divided condition of the charcoal, that they may be received on a handkerchief or the hand without burning them. The difficulty consists in effecting the complete mixture of the charcoal. The other two ingredients must first be thoroughly powdered separately, and again triturated when mixed, and finally the charcoal must be rubbed in carefully, till the whole is of a uniform tint of grey and very nearly black, and as the mixture proceeds portions must be rammed into a paper case, and set on fire; if the stars or pinks come out in clusters, and spread well without other and duller sparks, it is a sign that the whole is well mixed; but if the sparks are accompanied with dross, and are projected out sluggishly, and take some time to burn, the mixture and rubbing in the mortar must be continued; and even that must not be carried too far, or the sparks will be too small. N.B.—If the lamp-black was heated red hot in a close vessel, it would probably answer better when cold and powdered. Twelfth Experiment. Into a tall gas jar with a wide neck project some red-hot lamp-black through a tin funnel, when a most brilliant flame-like fire is obtained, showing that finely divided charcoal with pure oxygen would be sufficient to afford light; but as the atmosphere consists of oxygen diluted with nitrogen, compounds of charcoal with hydrogen, are the proper bodies to burn, to produce artificial light. Thirteenth Experiment. The Bude Light. This pretty light is obtained by passing a steady current of oxygen gas (escaping at a very low pressure) through and up the centre pipe of an argand oil lamp, which must be supplied with a highly carbonized oil and a very thick wick, as the oxygen has a tendency to burn away the cotton unless the oil is well supplied, and allowed to overflow the wick, as it does in the lamps of the lighthouses. The best whale oil is usually employed, though it would be worth while to test the value of Price's Belmontine Oil for the same purpose. (Fig. 102.)
- 86. Fig. 102. a. Reservoir of oil. b. The flexible pipe conveying oxygen to centre of the argand lamp. Fourteenth Experiment. A Red Light. Clear out the oil thoroughly from the Bude light apparatus; or, what is better, have two lamps, one for oil, and the other for spirit; fill the apparatus with a solution of nitrate of strontia and chloride of calcium in spirits of wine, and let it burn from the cotton in the same way as the oil, and supply it with oxygen gas. Fifteenth Experiment. A Green Light. Dissolve boracic acid and nitrate of baryta in spirits of wine, and supply the Bude lamp with this solution. Sixteenth Experiment. A Yellow Light. Dissolve common salt in spirits of wine, and burn it as already described in the Bude light apparatus. Seventeenth Experiment. The Oxy-calcium Light. This very convenient light is obtained in a simple manner, either by using a jet of oxygen as a blowpipe to project the flame of a spirit lamp on to a ball of lime; or common coal-gas is employed instead of the spirit lamp, being likewise urged against a ball of lime. By this plan one bag containing oxygen suffices for the production of a brilliant light, not equal, however, to the oxy-hydrogen light, which will be explained in the article on hydrogen. (Fig. 103.) Fig. 103. No. 1. a. Oxygen jet. b. The ball of lime, suspended by a wire. c. Spirit lamp. No. 2. d. Oxygen jet. e. Gas (jet connected with the gas-pipe in the rear by flexible pipe) projected on to ball of lime, f. Eighteenth Experiment. To show the weight of oxygen gas, and that it is heavier than air, the stoppers from two bottles containing it may be removed, one bottle may be left open for some time and then tested by a lighted taper, when it will still indicate the presence of the gas, whilst the other may be suddenly inverted over a little cup in which some ether, mixed with a few drops of turpentine, may be burning—the flame burns with much greater brilliancy at the moment when the oxygen comes in contact with it.
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