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.

Published by Dave

"May all beings be happy and peaceful. Thank you, Lord GOD! Praise you, Lord GOD! Love you, Lord GOD! Trust you, Lord GOD! Bless you, Lord GOD! Amen!"

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

<span>%d</span> bloggers like this: