How to find the primes that are part of Goldbach
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This document discusses a method for determining if Goldbach's conjecture is true for a given even number 2n. It presents a formula (formula 3) that checks if two prime numbers p and q exist such that their sum is 2n. The formula uses sine functions and multiplications to test all possible combinations of p and q between 2 and 2n-2. An example application of the formula in a computer program finds a solution, providing evidence for Goldbach's conjecture being true for that even number. The document concludes by stating the remaining step is a formal proof that the formula always produces a valid q.
How to find the primes that are part of Goldbach
- 1. How to determine Goldbach’s q’s where 2n=p+q The starting point is an earlier derived formula to check if a number is prime or not: ିଵ ܲሺ݊ሻ = ෑ sin ቀ (formula 1.) ୀଶ ߨ݊ ቁ <> 0 => ݊ = Prime ݅ Goldbach’s conjecture states that every even number can be written as the sum of two primes. We can agree on following convention: 2n q p every even number prime number greater than 2 (and q≤p) prime number smaller than 2n-2 With: 2n=p+q (formula 2.) By increasing q in formula 2, p will automatically decrease if 2n is fixed. Author : chrisdecorte@yahoo.com Page 1
- 2. We can now scan the range between 2 and 2n-2 for a suitable combination of q and p to find couples of (q,p) where Goldbach is valid for 2n. We can do this using the following new formula where we temporarily put q=x and p=2n-x: ௫ିଵ ሺଶି௫ሻିଵ ୀଶ ୀଶ ߨݔ ܩሺݔሻ = ෑ sin ቀ ቁ . ݅ ෑ ߨሺ2݊ − ݔሻ sin ൬ ൰ <> 0 => ݍ + = ݊2 ݀݊ܽ ݍ = ݔ ݆ (formula 3.) We tested the formula in PARI/GP using the following code (example for 2n=22): for(two_n=22,22,for(x=2,two_n,print(x," ", prod(i=2,x1,sin(Pi()*x/i),{i=1})*prod(j=2,two_n-x-1,sin(Pi()*(two_n-x)/j),{j=1})))) We found the following correct result: We now have to prove that there is always a valid q in formula 3 to prove Goldbach’s conjecture. Author : chrisdecorte@yahoo.com Page 2