# 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, $e > 4 * 10^{18}$, 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, $p_{i}$, less than e. Therefore, we generate a system of k Diophantine equations:

$e = 3 + n_{1}q_{1}$;

$e = 5 + n_{2}q_{2}$;

$e = 7 + n_{3}q_{3}$;

$e = p_{k} + n_{k}q_{k}$

where odd integer, $n_{i} > 1$ for $i \in$ {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.