![This “doubling” process can be iterated indefinitely. So, every C(x) in which x has a power of 2 in its
prime factors can be reduced by dividing it for the power of 2 that it contains.
Let's define “inverse sequence” the sequence obtained applying the reverse algorhythm, computing
the predecessor of an integer x.
Even if the normal sequences are univoke, the inverse sequences must take into account the fact that
some integers can have two predecessors (the reason of the name “crossing”).
Examining the inverse sequences we decide to choose always the minor of the two predecessors.
In this case the inverse function is defined below:
(n.b – per R intero = for R integer. “in tutti gli altri casi” = “in all the remaining cases”)
An example of inverse sequence that we call is the following:
11,22,7,14,28,9,18,36,72…..
Because a multiple of three multiplied by two is again a multiple of 3, we can say that when an inverse
sequence “encounter” a multiple of 3 it diverges to infinite.
It's likely that choosing the predecessors in a way to avoid the multiples of 3, we can generate
sequences of infinite length. Demonstrating this conjecture is equivalent to demonstrating the Collatz
conjecture.
The demonstration proposal
As underlined before, our demonstration proposal is not the main purpose of this document, but only
the starting point for the study on the extension of those recursive functions.
We have already verified that we don't need to demonstrate the conjecture for sequences with even
header. Now let's analyze the integers in form 3n, 3n+1 and 3n+2.
To have an odd 3n n must be odd, so in form 2k+1. By sibstitution we have 6k+3.
So its successor is [3(6k+3)]+1 = 18k+10 which is even. In our notation we'd say that C(6k+3,1)=18k+10
and C(6k+3,2)=9k+5
9k+5 can be even or odd depending on k, but in any case:
So it's in form 3Y+2](https://image.slidesharecdn.com/recursivefunctions-160309111421/85/Study-on-a-class-of-recursive-functions-4-320.jpg)
![In the same way, integers in form 3n+1 are even if n=2k so by substitution 6k+1.
C(6k+1,1)=18k+4 which is even, so C(6k+1,2)=9k+2 which is even or odd depending on k, but is
The sequences starting from the other subclasses encounter an integer in form 3n+2 in a very limited
number of steps, if we demonstrate the validity of the conjecture for this subclass, it will be
demonstrated for all the integers.
To have an even 3n+2 we need that n=2k+1, so by substitution 6k+5. We have conjectured that all the
integers in form 3n+2 (so 6k+5) are never primitives, so they appear in sequences with a minor header.
Let's call C(x,-1) the predecessor of x.
Let's consider C(6k+5,-1)=C(6k+5,0)*2 = 12k+10. In this case 12k+10-1=12k+9 which is divisible by 3, so
C(6k+5,-2) = [C(6k+5,-1)-1]/3 = 4k+3
We verify then that C(4k+3,1)=12k+10 consequently C(4k+3,2)=6k+5
So we can state that:
Every number in form 6k+5 appears in C(4k+3) and because 4k+3 < 6k+5 the conjecture is
demonstrated.
The g(n)
Before his famous conjecture, Collatz examined another recursive function, very much challenging,
whose definition is:
The sequences generated by this function differ fro those from Hailstone for at least three important
characteristics. We clearly state that those characteristics are conjectured, not demonstrated.
1) The inverse sequences are univoque. There are no corssings.
2) Every integer belongs to one and only one sequence.
3) Some sequences enter a loop. Other seem to diverge.](https://image.slidesharecdn.com/recursivefunctions-160309111421/85/Study-on-a-class-of-recursive-functions-5-320.jpg)
![Because every odd is followed by an even, we can compact the first division at every increment, so we
can rewite it as:
N.b. – [] means Round(), so the integer part rounded by defect.
In this notation, the definition of this function is similar to that of g(n), this lead us to think to the
possibility of extending the study to functions with a bigger divisor.
Extension to all the divisors: the Collatz-like functions
Let's call “base” the divisor (or module) of the function.
Let's call “increment” the case in which the successor is greater than the predecessor
Let's call “decrement” the case in which the successor is minor than the predecessor
In the definition of those finctions there are various parameters that have an impact on the sequences.
Let's call “standard functions” those in which the decrement of x is x-(x/base) and it's applied to the
case x = 0 (MOD base). Roundings are both by defect or by excess.
In the case of even bases we have an odd number of increments. As a “central” case doesn't exist we
arbitrarily round by defect when x = base/2 (MOD base)
To help in computations we have used a javascript program. To show the functions and their
parameters we will use the main screen of that program.
Some examples below:
This figure means:](https://image.slidesharecdn.com/recursivefunctions-160309111421/85/Study-on-a-class-of-recursive-functions-10-320.jpg)
Study on a class of recursive functions
- 1. Study on a class of recursive functions by Bruno Gentili and Luca Polidori Introduction The purpose of this document consists in showing the characteristics of a class of recursive functions which can be defined as extensions of the function known as “Hailstone Sequences”. We initially approached the Collatz conjecture trying to find a demonstration of the Collatz conjecture, but later this study has lead us to define a family of functions that we have called Collatz- like which we will detail later on. The demonstration proposal is included in this document but it's not its final scope. We think it's useful showing the path that we have followed and the further conjectures that we have enunciated about this family of functions and the sequences of integers that they generate. The Collatz conjecture Lothar Collatz (1910-1990) was a German mathematician. Even though he has published a number of important works, he is famous mainly for the conjecture named from him, also known as “Syracuse problem” or “3x + 1 Problem”. Let C(x) be a recursive function so that: N.b: “per” in Italian means “for”. I won't translate the pictures. Collatz conjectured that starting from any positive integer, all the generated sequences converge to 1, or better, they end in a loop 1,4,2,1.... This conjecture has been “brutally” verified with computers up to values around 10^50. Therefore, we decided to approach the conjecture trying a demonstration by induction. Below the fifth postulate of Peano-Dedekind, known as “axiom of induction”. ( 0 ∈ M, n ∈ M → n' ∈ M ) → M = N
- 2. In brief, if a proposition is true for x=1, and we verify that it's true for x+1, it's true for all the natural numbers. Analyzing the sequences, we can spot immediately that the sequences are fundamentally determined by the Module 6: - If an integer is in form 6n+1, the successor is in form 6n+4 - If an integer is in form 6n+2, the successor can be in form 6n+1 or 6n+4 - If an integer is in form 6n+3, the successor is in form 6n+4 - If an integer is in form 6n+4, the successor can be in form 6n+2 o 6n+5 - If an integer is in form 6n+5, the successor is in form 6n+4 - If an integer is in form 6n, the successor is in form 6n+3 At this point, we need to create an appropriate glossary in order to simplify the description. Let's start defining “header” the first element of a sequence. As example: C(11)=11,34,17,52,26,13,40,20,10,5,16,8,4,2,1….. The header is 11. It's obvious that if an integer x appears in the sequence of a header k < x, we don't need to demonstrate that C(x) converges to 1, because it's completely included into C(k). In the example above, C(17) is included into C(11). For the same reason, the even headers don't need to be demonstrated as the first operation is always a division by 2, so for x even C(x) is inclusive of C(x/2). Now it's important to understand whether there is a rule that determines whether a C(x) with x odd is included into another sequence or not. Let's define “primitive” an integer x so that doesn't exist a C(k) which contains x for k < x.
- 3. We noticed that the headers in form 3n+2 (or 6n+2 and 6n+5) are NEVER primitives. In particular, in base 10, the integers with numeric root 2,4,5 and 8 are never primitives. Integers with numeric root 1 and 7 are almost always primitives but with few rare exceptions which we have called “false primitives”. The first two false primitives are 61 and 91 which oth appear in C(27). False primitives are not very important. We will treat the integers with numeric root 1 and 7 as they are all primitives, so they need to be demonstrated. Finally, all the multiples of 3 are primitives. Let's now see what a sequence need to converge to 1, what's the neceesary and sufficient condition: Let's define C(k,n) the nth element of C(k). Let's define “predecessor of x”the integer k that C(k,n+1)=x. Let's define “Crossing” the integers x which has more than one possible predecessor. In order to converge to 1 a sequence must contain a predecessor of 4 (other than 1). In addition, because all the even numbers are divided by two at the following step, all the powers of 2 converge to 1. Let look to the table below: 1 2 4 8 16 3 2 64 128 256 512 102 4 2048 4096 8192 3x+1 0 1 5 21 85 341 1365 C(x,- 1) 0 0 10 42 170 682 2730 C(x,- 2) 0 0 3 84 340 227 5460 We see that even though (85-1)/3 is an integer, 29 is even so it can't be predecessor of 85 as it's successor is 14. One thing that we will see also in other functions of this class is that similar tables can be built by multiplying the first row by any even integer. 5 10 20 4 0 80 160 32 0 640 1280 2560 5120 10240 2048 0 40960 3x+1 3 1 3 53 213 853 3413 13653 C(x,- 1) 6 2 6 106 426 1706 6826 455 C(x,- 2) 12 5 2 35 852 3412 13650 910
- 4. This “doubling” process can be iterated indefinitely. So, every C(x) in which x has a power of 2 in its prime factors can be reduced by dividing it for the power of 2 that it contains. Let's define “inverse sequence” the sequence obtained applying the reverse algorhythm, computing the predecessor of an integer x. Even if the normal sequences are univoke, the inverse sequences must take into account the fact that some integers can have two predecessors (the reason of the name “crossing”). Examining the inverse sequences we decide to choose always the minor of the two predecessors. In this case the inverse function is defined below: (n.b – per R intero = for R integer. “in tutti gli altri casi” = “in all the remaining cases”) An example of inverse sequence that we call is the following: 11,22,7,14,28,9,18,36,72….. Because a multiple of three multiplied by two is again a multiple of 3, we can say that when an inverse sequence “encounter” a multiple of 3 it diverges to infinite. It's likely that choosing the predecessors in a way to avoid the multiples of 3, we can generate sequences of infinite length. Demonstrating this conjecture is equivalent to demonstrating the Collatz conjecture. The demonstration proposal As underlined before, our demonstration proposal is not the main purpose of this document, but only the starting point for the study on the extension of those recursive functions. We have already verified that we don't need to demonstrate the conjecture for sequences with even header. Now let's analyze the integers in form 3n, 3n+1 and 3n+2. To have an odd 3n n must be odd, so in form 2k+1. By sibstitution we have 6k+3. So its successor is [3(6k+3)]+1 = 18k+10 which is even. In our notation we'd say that C(6k+3,1)=18k+10 and C(6k+3,2)=9k+5 9k+5 can be even or odd depending on k, but in any case: So it's in form 3Y+2
- 5. In the same way, integers in form 3n+1 are even if n=2k so by substitution 6k+1. C(6k+1,1)=18k+4 which is even, so C(6k+1,2)=9k+2 which is even or odd depending on k, but is The sequences starting from the other subclasses encounter an integer in form 3n+2 in a very limited number of steps, if we demonstrate the validity of the conjecture for this subclass, it will be demonstrated for all the integers. To have an even 3n+2 we need that n=2k+1, so by substitution 6k+5. We have conjectured that all the integers in form 3n+2 (so 6k+5) are never primitives, so they appear in sequences with a minor header. Let's call C(x,-1) the predecessor of x. Let's consider C(6k+5,-1)=C(6k+5,0)*2 = 12k+10. In this case 12k+10-1=12k+9 which is divisible by 3, so C(6k+5,-2) = [C(6k+5,-1)-1]/3 = 4k+3 We verify then that C(4k+3,1)=12k+10 consequently C(4k+3,2)=6k+5 So we can state that: Every number in form 6k+5 appears in C(4k+3) and because 4k+3 < 6k+5 the conjecture is demonstrated. The g(n) Before his famous conjecture, Collatz examined another recursive function, very much challenging, whose definition is: The sequences generated by this function differ fro those from Hailstone for at least three important characteristics. We clearly state that those characteristics are conjectured, not demonstrated. 1) The inverse sequences are univoque. There are no corssings. 2) Every integer belongs to one and only one sequence. 3) Some sequences enter a loop. Other seem to diverge.
- 6. Let's give a look to some of those sequences: 1 1 1 LOOP 2 2 3 2 LOOP 4 4 5 7 9 6 4 LOOP But mainly: 8 11 15 10 13 17 23 31 41 55 73 97 129 86 115 153 102 68 91 121 161 215 287 383 511 681 454 605 807 538 717 478 637 849 566 755 1007 1343 1791 1194 796 1061 1415 1887 1258 1677 1118 1491 994 1325 1767 1178 1571 2095 2793 1862 2483 3311 4415 5887 7849 10465 13953 9302 12403 16537 22049 29399 39199 52265 69687 46458 30972 20648 27531 18354 12236 16315 21753 14502 9668 12891 8594 11459 15279 10186 13581 9054 6036 4024 5365 7153 9537 6358 8477 11303 15071 20095 26793 17862 11908 15877 21169 28225 37633 50177 66903 44602 59469 39646 52861 70481 93975 62650 83533 111377 148503 99002 132003 88002 58668 39112 52149 34766 46355 61807 82409 109879 146505 97670 130227 86818 115757 154343 205791 137194 182925 121950 81300 54200 72267 48178 64237 85649 114199 152265 101510 135347 180463 240617 320823 213882 142588 190117 253489 337985 450647 600863 801151 1068201 712134 474756 316504 422005 562673 750231 500154 333436 444581 592775 790367 1053823 1405097 1873463 2497951 3330601 4440801 2960534 3947379 2631586 3508781 4678375 6237833 8317111 11089481 14785975 19714633 26286177 17524118 23365491 15576994 20769325 27692433 18461622 12307748 16410331 21880441 29173921 38898561 25932374 34576499 46101999 30734666 40979555 54639407 72852543 48568362 32378908 43171877 57562503 38375002 51166669 68222225 90962967 60641978 80855971 107807961 71871974 95829299 127772399 170363199 113575466 151433955 100955970 67303980 44869320 29912880 19941920 26589227 35452303 47269737 31513158 21008772 14005848 9337232 12449643 8299762 11066349 7377566 9836755 13115673 8743782 5829188 7772251 10363001 13817335 18423113 24564151 32752201 43669601 58226135 77634847 103513129 138017505 92011670 122682227 163576303 218101737 145401158 193868211 129245474 172327299 114884866 153179821 204239761 272319681 181546454 242061939 161374626 107583084 71722056 47814704 63752939 85003919 113338559 151118079 100745386 134327181 89551454 119401939 159202585 212270113 283026817 377369089 503158785 335439190 447252253 596336337 397557558 265038372 176692248 117794832 78529888 104706517 139608689 186144919 248193225 165462150 110308100 147077467 196103289 130735526 174314035 232418713 309891617 413188823 550918431 367278954 244852636 326470181 435293575 580391433 386927622 257951748 171967832 229290443 305720591 407627455 543503273 724671031 966228041 1288304055 858869370 572579580 381719720 508959627 339306418 452408557 603211409 804281879 1072375839 714917226 476611484 635481979 847309305 564872870 753163827 502109218 669478957 892638609 595092406 793456541 1057942055 1410589407 940392938 1253857251 835904834 1114539779 1486053039 990702026 1320936035 1761248047 2348330729 3131107639 4174810185 2783206790 3710942387 4947923183 6597230911 8796307881 5864205254 7818940339 10425253785 6950169190 9266892253 12355856337 8237237558 10982983411 14643977881 19525303841 26033738455 34711651273 46282201697 61709602263 41139734842 54852979789 73137306385 97516408513 130021878017 173362504023 115575002682 77050001788 102733335717 68488890478 91318520637 60879013758 40586009172 27057339448 36076452597 24050968398 16033978932 10689319288 14252425717 19003234289 25337645719 33783527625 22522351750 30029802333 20019868222 26693157629 35590876839 23727251226 15818167484 21090889979 28121186639 37494915519 24996610346 33328813795 44438418393 29625612262 39500816349 26333877566 35111836755 23407891170 15605260780 20807014373 27742685831 36990247775 49320330367 65760440489 87680587319 116907449759 155876599679
- 7. 207835466239 277113954985 369485273313 246323515542 164215677028 218954236037 291938981383 389251975177 519002633569 692003511425 922671348567 615114232378 820152309837 546768206558 729024275411 972032367215 1296043156287 864028770858 576019180572 384012787048 512017049397 341344699598 455126266131 303417510754 404556681005 539408908007 719211877343 958949169791 1278598893055 1704798524073 1136532349382 1515376465843 2020501954457 2694002605943 3592003474591 4789337966121 3192891977414 4257189303219 2838126202146 1892084134764 1261389423176 1681852564235 2242470085647 1494980057098 1993306742797 2657742323729 3543656431639 4724875242185 6299833656247 8399778208329 5599852138886 7466469518515 9955292691353 13273723588471 17698298117961 11798865411974 15731820549299 20975760732399 13983840488266 18645120651021 12430080434014 16573440578685 11048960385790 14731947181053 9821298120702 6547532080468 8730042773957 11640057031943 15520076042591 20693434723455 13795623148970 18394164198627 12262776132418 16350368176557 10900245451038 7266830300692 9689107067589 6459404711726 8612539615635 5741693077090 7655590769453 10207454359271 13609939145695 18146585527593 12097723685062 16130298246749 21507064328999 28676085771999 19117390514666 25489854019555 33986472026073 22657648017382 30210197356509 20140131571006 26853508761341 35804678348455 47739571131273 31826380754182 42435174338909 56580232451879 75440309935839 50293539957226 67058053276301 89410737701735 119214316935647 158952422580863 211936563441151 282582084588201 188388056392134 125592037594756 167456050126341 111637366750894 148849822334525 198466429779367 264621906372489 176414604248326 235219472331101 313625963108135 418167950810847 278778633873898 371704845165197 495606460220263 660808613627017 881078151502689 587385434335126 783180579113501 1044240772151335 1392321029535113 1856428039380151 2475237385840201 3300316514453601 2200211009635734 1466807339757156 977871559838104 1303828746450805 1738438328601073 2317917771468097 3090557028624129 2060371352416086 1373580901610724 915720601073816 1220960801431755 813973867621170 542649245080780 723532326774373 964709769032497 1286279692043329 1715039589391105 2286719452521473 3048959270028631 4065279026704841 5420372035606455 3613581357070970 4818108476094627 3212072317396418 4282763089861891 5710350786482521 7613801048643361 10151734731524480 ………………….. The sequence g(14) seems to diverge, too. An important notice is: 44 44 59 79 105 70 93 62 83 111 74 99 66 44 LOOP Analyzing the inverse sequences it looks like they diverge, too, but more slowly: 8 12 18 27 20 30 45 34 51 38 57 43 32 48 72 108 162 243 182 273 205 154 231 173 130 195 146 219 164 246 369 277 208 312 468 702 1053 790 1185 889 667 500 750 1125 844 1266 1899 1424 2136 3204 4806 7209 5407 4055 3041 2281 1711 1283 962 1443 1082 1623 1217 913 685 514 771 578 867 650 975 731 548 822 1233 925 694 1041 781 586 879 659 494 741 556 834 1251 938 1407 1055 791 593
- 8. 445 334 501 376 564 846 1269 952 1428 2142 3213 2410 3615 2711 2033 1525 1144 1716 2574 3861 2896 4344 6516 9774 14661 10996 16494 24741 18556 27834 41751 31313 23485 17614 26421 19816 29724 44586 66879 50159 37619 28214 42321 31741 23806 35709 26782 40173 30130 45195 33896 50844 76266 114399 85799 64349 48262 72393 54295 40721 30541 22906 34359 25769 19327 14495 10871 8153 6115 4586 6879 5159 3869 2902 4353 3265 2449 1837 1378 2067 1550 2325 1744 2616 3924 5886 8829 6622 9933 7450 1117 8381 6286 9429 7072 10608 15912 23868 35802 53703 40277 30208 45312 67968 101952 152928 229392 344088 516132 774198 1161297 870973 653230 979845 734884 1102326 1653489 1240117 930088 1395132 2092698 3139047 2354285 1765714 2648571 1986428 2979642 4469463 3352097 2514073 1885555 1414166 2121249 …….. As well as for the Hailstone function, the divisibility by the base is fundamental. In example: generalized to n.b - “passaggi” = “steps” In the attempt to understant if the non-looping sequences are effectively divergent, we have hypotized that 8 and 14 are the minimum values of two infinite sequences. If effectively no crossings exist in both the directions, this is the only possibility because all the loops of this function are closed. We mean, a sequence can't contain a loop. It can only be completely part of the loop as in the cases of g(4) and g(44). Below the table of the first “minimums” identified:
- 9. We see that: - All the minimums are even - They are mainly in form 4n (with few exceptions) We must take into account that with the short extension of our calculations and the little computing capacity of normal windows Pcs, we can't verify if some of those apparently distinct sequences are not part of the same one. The reason why the minimums are even is trivial: So it's obvious that the minimum of a sequence must have 2 in its factors. After a number of attempts and alternative approaches, we thought to extend this class of function to cases with more than 3 choices. Let's look back to the Hailstone function:
- 10. Because every odd is followed by an even, we can compact the first division at every increment, so we can rewite it as: N.b. – [] means Round(), so the integer part rounded by defect. In this notation, the definition of this function is similar to that of g(n), this lead us to think to the possibility of extending the study to functions with a bigger divisor. Extension to all the divisors: the Collatz-like functions Let's call “base” the divisor (or module) of the function. Let's call “increment” the case in which the successor is greater than the predecessor Let's call “decrement” the case in which the successor is minor than the predecessor In the definition of those finctions there are various parameters that have an impact on the sequences. Let's call “standard functions” those in which the decrement of x is x-(x/base) and it's applied to the case x = 0 (MOD base). Roundings are both by defect or by excess. In the case of even bases we have an odd number of increments. As a “central” case doesn't exist we arbitrarily round by defect when x = base/2 (MOD base) To help in computations we have used a javascript program. To show the functions and their parameters we will use the main screen of that program. Some examples below: This figure means:
- 11. Since now on we will abamdon the algebraic notation beacsue we think that using the program main screen is simpler, as in the case of base 11 below: We can see that in the cases with odd base, the roundings by defect are as many as the roundings by excess. Conjectures on Collatz-like functions The first thing that appears evident is that all the standard sequences except Hailstone seem to diverge, or at least they contain looping sequences analogous to the g(n). Unlike the g(n), in the standard function in base 5 we have crossings. This means that different sequences can “collapse” in the same loop. This is a consequence of thr roundings: with more cases available, more integers can be rounded to the same. In the case above, both 6 and 7 have increment 1.
- 12. It's interesting to notice that the crossings in the sequences from 1 to 100 are: 4 - 8 - 16 - 20 - 28 - 32 - 40 - 44 - 52 - 56 - 64 - 68 - 76 – 80 We also see that there are only 4 loops: 1 2 3,4 6,8 Let's call 1-loop the loops of only 1 repeated element. Let's call 2-loop the loops of two elements. All the other loops are called “extemporary” because they don't follow an apparent rule. Analyzing the functuins with greater bases we have deduced that: In the standard functions the 1-loops are round(base/2) and go from 1 to round(base/2). 2-loops are round(base/2) and are: base-1,base 2*(base-1),2*base Round(base/2)*(base-1),Round(base/2)*base If we go back to g(n) we see that it effectively has a 1-loop (1) and a 2-loop (2,3). If we consider Hailstone with the first division collapsed we have 0 1-loops and 1 2-loop (1,2). This confirms the formula and demonstrates that, unless there are extemporary loops very “high” (over 10^50), the non-divergent sequences in Hailstone can converge only to 1. Also the crossings seem to follow a precise rule: Base Crossings 2 1,4,7,10,13,16… 3 No 4 6,9,12,21,24,27,33… 5 4,8,16,20,28,32,40,44,52,56… 6 5,10,15,25,30,35,45,50,55… 7 6,18,30,42,54,66…. 8 7,21,28,35,42,49,56,70,77,84,91,98,105,112,119,133 … 9 8,16,24,32,48,56,64,72,88… 10 9,18,27,36,45,54,63,81,90,108,117,126,135,144… 11 10,20,40,50,70,80,100…
- 13. It appears an evident regularity. Only on base 3 standard, the g(n) seems to be an exception. We'll see later that it is not. We notice that with al the other bases the first crossing is “base – 1” unless it's coincident with a 2-loop as it is in base 4. Also the “holes” in the sequences of crossings follow a schema but we still have to investigate the algorhithm that produces them. We see that: base 4 – crossings multiple of 3 – a hole every 3 multiples base 5 – crossings multiple of 4 – a hole every 3 multiples base 6 – crossings multiple of 5 – a hole every 4 multiples base 7 – crossings multiple of 6 – a hole every 2 multiples base 8 – crossings multiple of 7 – a hole every 7 multiples base 9 – crossings multiple of 8 – a hole every 5 multiples It doesn't seem to exist a direct algorhitmic relation between base Non sembra esserci una relazione algoritmica diretta fra base e “holes in the crossings”. We have a number of parameters that we can use in defining those functions: we can decide to vary the increments and the decrements. We notice that the cenvergence or divergence of the sequences is depending only on those parameters. In example, if in Hailstone we increase the increment, the sequences grow more, but they continue to converge, even if we can generate a number of extemporary loops. See the example below. The sequences shown are just some of the extemporary loops “captured” by our program.
- 14. Things are very different if, insetad of increasing the increment or the decrement, we apply a multiplicative coefficient. NON-DIVERGENCE CONJECTURE: The sequences donìt diverge if a coefficient >= base/2 is applied to the decrement. Demonstrating this conjecture automatically implies the demonstration of the Collatz conjecture because in Hailstone the base is 2 and the decrement has coefficient 1. This woukd also demonstrate the non-convergence of g(n) because the ratio coefficient/base is 1/3 < ½. Applying a coefficient 2 to the decrement in g(n) we see that - All the sequences seem to converge rapidly - There is only 1 1-loop (1) - There aren't extemporary loops. - Crossings appear (1,3,5,7,9,11….) and they don't have “holes”.
- 15. Another example randomly chosen: base 19 with coefficient 10:
- 16. We don't have a demonstration. CONJECTURE ON THE ABSENCE OF CROSSINGS Base/2 seems to be the limit value of the coefficient for the non-divergence. It means that if the coefficient is minor than base/2, the functions contain divergent sequences. Anyway, increments and decrements can be manipulated in order to obtain functions without crossings as in g(n).
- 17. When the coefficient of the decrement is “round(base/2) -1” the functions don't have crossings, but only if the base is odd. This limitation on the base may depend on the asymmetry of the roundings on the increments. We still have to analyze this possibility. The table below shows some functions with no crossings. base Coefficient 3 1 (g(n)) 5 2 7 3 9 4 11 5 13 6 … … In some cases we can obtain function without crossings also modifying the increments, i.e. Applying a coefficient 5 to the increments of the g(n) as in the example below: The table below shows all the functions without crossings with a coefficient on the increment: base R1 R2 R3 R4 3 5 5 3 7 7 3 13 13 5 1 3 3 1 5 3 7 7 3 5 3 3 3 3 It's likely that other combinations of coefficients on the increments can produce sequences without crossings, but it's significant that we havan't found any after base 5.
- 18. EXTENSION TO NEGATIVE NUMBERS Until now, we didn't make any distinction between the remainder of a division and the concept of congrience in modular arithmetic. If we want to extend the class of functions to the domain of negative integers we must think in terms of congruences. I.e. 23 ≡ 9 (MOD 14) but -23 ≡ 5 (MOD 14). Our software doesn't work with negative sequences, but we can produce the absolute values just inverting the signs of increments and decrements. Below, the negative versions of some functions. You should imagine the minus sign in front of every integer. It's noticeable as the negative version of Hailstone which is in base two is so similar to g(n) which is in base 3.
- 19. In the case above we have two decrements and one increment. This should be equivalent to a mutiplicative coefficient and effectively the sequences converge.
- 20. As well as we can obtain the equivalent of the negative sequences or theabsence of crossings, we can obtain a wide range of effects just varying some parameters: 1) Variation of the position of the decrement We have seen that moving the decrement, which consists in inverting the signs of the decrement and one of the increments, doesn't cause a variation to the general behaviour of the sequences. 2) Variation of the sign of the roundings Other than inverting all the decrements and all the increments, we can decide to go for intermediate situations and position changes. In any case the “base/2 rule” still seems to be valid. In those cases there's an interesting fact about loops. We have loops with more than 2 elements which can't be considered extemporary. 3) Multiplicative coefficients on both decrements and increments As in case 2) those variations cause changes in the loops and the base/2 conjecture still seems valid.
- 21. All those considerations must take into account the insufficient computing capacity and the architectural limits of javascript. We see that even applying to an increment the same coefficient of the decrement, the non-divergence of the sequences seems to depend only on the decrement. Effectively, also the position of the increment with the coefficient seems to be relevant: In the case below we don't have divergences in the sequences from 1 to 1000 and all of them are almost short. In this case, instead, we have sequences growing fast which may diverge: As in this case; the limit MAX is inserted by the program when the architectural limit of javascript on the number of digits is reached. 44 53 64 77 122 194 233 280 112 178 214 257 410 164 197 314 377 602 962 1538 1846 2215 886 1063 1276 1531 1837 2938 3526 4231 5077 8122 12994 15593 18712 29938 35926 43111 51733 62080 24832 39730 15892 25426 30511 36613 43936 52723 63268 75922 121474 145769 174923 209908 251890 100756 120907 193450 77380 30952 49522 79234 95081 114097 182554 219065 87626 105151 126181 151417 242266 290719 348863 418636 502363 602836 723403 868084 1041701 1250041 1500049 1800059 2160071 2592085 1036834 1244201 1493041 1791649 2149979 2579975 1031990 412796 495355 198142 317026 380431 456517 730426 876511 1051813 1262176 1514611 1817533 2181040 872416 1046899 1256279 1507535 603014 723617 1157786 1389343 1667212 2667538 3201046 3841255 1536502 2458402 3933442 6293506 7552207 12083530 4833412 7733458 9280150 3712060 1484824 1781789 2138147 3421034 4105241 4926289 5911547 9458474 11350169 13620203 16344244 19613093 23535712 37657138 45188566 54226279 65071535 26028614 31234337 49974938 59969926 71963911 86356693 103628032 165804850 66321940 26528776 31834531 38201437 61122298 73346758 88016110 35206444 42247733 50697280 20278912 32446258 38935510 15574204 18689045 7475618 8970742 14353186 17223823 20668588 24802306 29762767 47620426 57144511 68573413 82288096 98745715 39498286 47397943 56877532 91004050 36401620 14560648 17472778 20967334 25160801 30192961 36231553 43477864 52173437 83477498 100172998 120207598 144249118 173098942 276958306 332349967 531759946 638111935 255244774 306293729 367552475 147020990 58808396 70570075 28228030 11291212 18065938 21679126 26014951 31217941 37461529 44953835 17981534 21577841 25893409 31072091 37286509 44743811 53692573 64431088 77317306 92780767 148449226 178139071 213766885 85506754 102608105 41043242 65669186 78803023 94563628 113476354 136171625 54468650 21787460 8714984 10457981 12549577 20079322 32126914 38552297 61683674 74020409 88824491 106589389 127907267 204651626 245581951 294698341 353638009 424365611 509238733
- 22. 611086480 244434592 391095346 469314415 187725766 225270919 270325103 324390124 389268149 467121779 560546135 224218454 269062145 107624858 129149830 51659932 82655890 33062356 39674827 63479722 101567554 121881065 48752426 58502911 70203493 84244192 134790706 161748847 258798154 310557785 124223114 149067737 238508378 286210054 343452065 137380826 164856991 197828389 237394067 379830506 455796607 729274570 291709828 350051794 420062153 504074584 604889501 725867401 871040881 1045249057 1672398490 668959396 802751275 321100510 128440204 154128245 61651298 73981558 88777870 35511148 42613378 51136054 61363265 24545306 29454367 47126986 56552383 67862860 27145144 32574173 39089008 46906810 18762724 22515269 27018323 32421988 38906386 46687663 56025196 67230235 26892094 32270513 38724616 46469539 55763447 89221514 107065817 171305306 205566367 328906186 394687423 473624908 568349890 227339956 272807947 436492714 523791257 838066010 335226404 402271685 160908674 193090409 231708491 278050189 333660227 533856362 854170178 1025004214 1230005057 1968008090 787203236 944643883 1133572660 453429064 544114877 870583802 1392934082 2228694530 891477812 1426364498 1711637398 2053964878 2464757854 2957709425 1183083770 473233508 567880210 227152084 272582501 327099001 392518801 471022561 565227073 678272488 813926986 976712383 1172054860 468821944 562586333 675103600 270041440 108016576 129619891 155543869 186652643 223983172 358373074 430047689 516057227 825691562 1321106498 1585327798 1902393358 2282872030 913148812 1461038098 1753245718 2103894862 3366231778 4039478134 4847373761 5816848513 6980218216 8376261859 10051514231 12061817077 19298907322 30878251714 37053902057 59286243290 23714497316 28457396779 34148876135 13659550454 16391460545 6556584218 7867901062 12588641698 15106370038 18127644046 21753172855 8701269142 13922030626 16706436751 20047724101 24057268921 28868722705 11547489082 18475982530 7390393012 11824628818 14189554582 22703287330 9081314932 14530103890 5812041556 6974449867 11159119786 13390943743 16069132492 25710611986 30852734383 37023281260 14809312504 17771175005 7108470002 11373552002 18197683202 29116293122 46586068994 55903282793 67083939352 107334302962 171734884738 206081861686 247298234023 296757880828 356109456994 427331348393 512797618072 820476188914 984571426697 1575314282714 1890377139257 3024603422810 1209841369124 1451809642949 1742171571539 2090605885847 3344969417354 4013963300825 1605585320330 642234128132 1027574605010 411029842004 493235810405 197294324162 315670918658 378805102390 151522040956 181826449147 290922318634 349106782361 418928138833 502713766600 201085506640 80434202656 96521043187 154433669098 185320402918 222384483502 355815173602 569304277762 910886844418 1093064213302 1748902741282 2798244386050 1119297754420 447719101768 537262922122 859620675394 1031544810473 1237853772568 1485424527082 2376679243330 950671697332 1521074715730 608429886292 973487818066 1168185381679 1401822458015 560728983206 672874779847 1076599647754 1291919577305 516767830922 826828529474 992194235369 1190633082443 1428759698932 2286015518290 914406207316 1097287448779 1316744938535 526697975414 632037570497 1011260112794 1213512135353 1456214562424 1747457474909 2096948969891 2516338763869 3019606516643 3623527819972 5797644511954 6957173414345 2782869365738 3339443238886 4007331886663 4808798263996 5770557916795 2308223166718 2769867800062 4431788480098 5318146176118 6381775411342 10210840658146 12253008789775 4901203515910 1960481406364 2352577687637 3764124300218 4516949160262 7227118656418 8672542387702 13876067820322 22201708512514 26642050215017 42627280344026 51152736412831 61383283695397 98213253912634 117855904695161 141427085634193 169712502761032 271540004417650 108616001767060 43446400706824 52135680848189 62562817017827 100100507228522 160160811565634 192192973878761 230631568654513 276757882385416 332109458862499 398531350634999 478237620761999 573885144914399 688662173897279 826394608676735 330557843470694 396669412164833 476003294597800 190401317839120 76160527135648 91392632562778 109671159075334 131605390890401 157926469068481 189511762882177
- 23. 303218820611482 485150112978370 194060045191348 232872054229618 279446465075542 447114344120866 536537212945039 643844655534047 1030151448854474 1236181738625369 1483418086350443 1780101703620532 2848162725792850 1139265090317140 455706036126856 546847243352227 874955589363562 1399928942981698 1679914731578038 2015897677893646 2419077213472375 967630885388950 387052354155580 154820941662232 247713506659570 99085402663828 118902483196594 142682979835913 171219575803096 205463490963715 82185396385486 98622475662583 118346970795100 47338788318040 18935515327216 22722618392659 27267142071191 32720570485429 39264684582515 15705873833006 18847048599607 30155277759370 12062111103748 14474533324498 17369439989398 20843327987278 25011993584734 30014392301681 36017270762017 57627633219226 69153159863071 82983791835685 33193516734274 39832220081129 47798664097355 19119465638942 30591145022306 36709374026767 58734998442826 70481998131391 84578397757669 101494077309203 121792892771044 146151471325253 175381765590304 210458118708365 84183247483346 101019896980015 40407958792006 48489550550407 77583280880650 31033312352260 12413324940904 14895989929085 5958395971634 7150075165961 8580090199153 10296108238984 12355329886781 14826395864137 23722233382618 28466680059142 45546688094626 54656025713551 65587230856261 78704677027513 94445612433016 113334734919619 136001681903543 163202018284252 261123229254802 417797166807682 668475466892290 267390186756916 320868224108299 385041868929959 462050242715951 554460291259141 665352349510969 798422819413163 958107383295796 1149728859954955 459891543981982 735826470371170 294330588148468 353196705778162 565114729245058 678137675094070 271255070037628 325506084045154 390607300854185 156242920341674 187491504410009 224989805292011 269987766350413 323985319620496 388782383544595 155512953417838 186615544101406 223938652921687 358301844674698 429962213609638 515954656331566 619145587597879 742974705117455 297189882046982 475503811275170 190201524510068 228241829412082 365186927059330 146074770823732 233719633317970 93487853327188 112185423992626 134622508791151 161547010549381 193856412659257 310170260254810 124068104101924 148881724922309 178658069906771 214389683888125 85755873555250 34302349422100 13720939768840 5488375907536 6586051089043 7903261306852 12645218090962 20232348945538 24278818734646 29134582481575 11653832992630 4661533197052 7458453115282 11933524984450 4773409993780 1909363997512 3054982396018 3665978875222 5865566200354 7038679440425 2815471776170 1126188710468 1351426452562 2162282324098 2594738788918 3113686546702 4981898474722 7971037559554 9565245071465 3826098028586 4591317634303 5509581161164 6611497393397 10578395829434 12694074995321 15232889994385 6093155997754 7311787197305 2924714878922 4679543806274 5615452567529 6738543081035 2695417232414 3234500678897 5175201086234 6210241303481 7452289564177 11923663302682 19077861284290 7631144513716 9157373416459 10988848099751 13186617719701 15823941263641 18988729516369 22786475419643 27343770503572 43750032805714 52500039366857 84000062986970 33600025194788 40320030233746 48384036280495 19353614512198 23224337414638 27869204897566 33443045877079 40131655052495 16052662020998 19263194425198 23115833310238 27738999972286 33286799966743 39944159960092 63910655936146 76692787123375 30677114849350 12270845939740 4908338375896 5890006051075 2356002420430 942400968172 1507841549074 1809409858889 2171291830667 3474066929066 4168880314879 5002656377855 2001062551142 3201700081826 3842040098191 4610448117829 5532537741395 2213015096558 2655618115870 1062247246348 1274696695618 1529636034742 2447417655586 2936901186703 3524281424044 4229137708853 5074965250624 6089958300749 7307949960899 8769539953079 10523447943695 4209379177478 5051255012974 6061506015569 7273807218683 8728568662420 3491427464968 4189712957962 6703540732738 8044248879286 9653098655143 11583718386172 18533949417874 22240739301449 26688887161739 32026664594087 51242663350538 61491196020646 73789435224775 29515774089910 11806309635964 14167571563157
- 24. 22668114501050 9067245800420 3626898320168 4352277984202 6963644774722 11141831639554 13370197967465 5348079186986 6417695024383 7701234029260 3080493611704 3696592334045 1478636933618 1774364320342 2838982912546 3406779495055 1362711798022 2180338876834 2616406652201 3139687982641 3767625579169 4521150695003 5425380834004 6510457000805 2604182800322 4166692480514 5000030976617 8000049562586 9600059475103 11520071370124 13824085644149 16588902772979 19906683327575 7962673331030 3185069332412 5096110931858 6115333118230 2446133247292 3913813195666 4696575834799 5635891001759 6763069202111 8115683042533 9738819651040 3895527860416 4674633432499 5609560118999 6731472142799 8077766571359 9693319885631 11631983862757 18611174180410 7444469672164 8933363606597 14293381770554 17152058124665 6860823249866 8232987899839 9879585479807 15807336767690 6322934707076 7587521648491 9105025978189 10926031173827 17481649878122 27970639804994 33564767765993 40277721319192 64444354110706 77333224932847 123733159892554 148479791871065 59391916748426 71270300098111 85524360117733 102629232141280 41051692856512 65682708570418 78819250284502 126110800455202 201777280728322 322843649165314 387412378998377 619859806397402 991775690235842 1586841104377346 1904209325252815 761683730101126 914020476121351 1096824571345621 1316189485614745 526475794245898 631770953095078 758125143714094 909750172456913 1091700206948296 1310040248337955 524016099335182 838425758936290 335370303574516 402444364289419 482933237147303 579519884576764 695423861492117 1112678178387386 1335213814064863 1602256576877836 1922707892253403 2307249470704084 2768699364844901 3322439237813881 3986927085376657 6379083336602650 2551633334641060 1020653333856424 1224784000627709 1469740800753251 1763688960903901 2116426753084681 2539712103701617 4063539365922586 4876247239107103 5851496686928524 7021796024314229 8426155229177075 3370462091670830 1348184836668332 2157095738669330 862838295467732 1380541272748370 552216509099348 662659810919218 795191773103062 1272306836964898 1526768204357878 1832121845229454 2198546214275345 879418485710138 1055302182852166 1266362619422599 1519635143307119 1823562171968543 2188274606362252 3501239370179602 5601982992287362 8963172787659778 10755807345191734 MAX Inverse functions The inverse functions have a particular significance if applied to functions with no crossings. Identifying a rule for their construption may help in demonstrating the divergence of those functions. We have seen before that if we want to obtain functions with odd base without crossings, we must apply a coefficient in the case n ≡0 (MOD base) by a coefficient k=n/(n+1) with base=n+2. We have verified that the inverse function is a function with even base b=(base + 1) where the cases n ≡ ≡0 (MOD base) and n base/2 (MOD base) have increment n/(2n+1) while the other cases have the sign inverted so they become decrements. Note: base indicates the odd base, not the base of the inverse function which is b. So we can say that starting from base=3, increasing the base the decrements are 1/3, 2/5, 3/7, 4/9... while for the even inverted bases the ratio is ½, 2/3, ¾, 4/5....
- 25. If we consider the even based sequences as inverse of the odd ones we see that for the g(n) the growth on the right is averagely +2n/3 while on the left is +n/2. This explains why the sequences grow faster to the right respect of to the left. Minimums of functions without crossings and odd base We have verified that the minimum values of the sequences with no crossings are multiples of (base – decrement coefficient): BASE Coeff. Incr. Minimum Decrement sx Increment dx 3 1 2n 1/3 1/2 5 2 3n 2/5 2/3 7 3 4n 3/7 3/4 9 4 5n 4/9 4/5 11 5 6n 5/11 5/6 13 6 7n 6/13 6/7 15 7 8n 7/15 7/8 Considerations Which utility can those functions have? Currently we have no idea, if they'll ever have one. But often in matehmatics and especially in number theory, apparently useless objects find applications in other fields as for example the matrixes in quantum mechanics. From a “phylosophical” point of view, we see an analogy between those sequences and the iterative processes used for divisions and roots. We can imagine the sequences as the non integer part of a division represented in an infinite numeric base. Non-convergent sequences are irrationals and/or transcendents, while extemporary loops may be intended as the equivalent of periodic numbers. Also 1-loops and 2-loops could be the equivament of the 0 in the non-periodic fractions. Just phylosophy, anyway. In any case it's evident that those functions are a valid clue to consider “true” the Collatz conjecture. Demostrating just one of the conjectures on those functions would demonstrate the Collatz conjecture, and in case our demonstration proposal has no flaws, we should have a good starting point to demonstrate all the other conjectures, including the non-conmvergence of the g(n).