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! 🙂


Published by Dave

"May all beings be happy and peaceful. Thank you, Lord GOD! Praise you, Lord GOD! Amen!" :-)

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