What could be an indirect proof of Goldbach’s Conjecture?

We can attempt a proof of Goldbach’s Conjecture via a proof by contradiction.

Suppose there exists a positive even integer, , that is not a sum of two odd primes.

We let e = p + nq (or equivalently, e ≡ p mod q) where odd primes, p and q, are less than e with odd integer, n > 1.

But there are k odd primes, , less than e. Therefore, we generate a system of k Diophantine equations:

;

;

;

…

where odd integer, for {1, 2, 3,…, k}.

Remark: We must show that least one of those equations is false (contradicting our assumption that e is not a sum of two odd primes) to prove Goldbach’s Conjecture is true.

Claim: We strongly believe Goldbach’s Conjecture is provable with some more analysis and with the help of some tools/methods from prime number theory.

A Crucial Idea: The distribution of all odd primes less than e helps us to prove Goldbach’s Conjecture.

Good Luck! 🙂

Dave.