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.

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
ሺଶ௡ି௫ሻିଵ
௜ୀ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