Goldbach Conjecture implies Weak Goldbach Conjecture

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Theorem

The Goldbach Conjecture:

Every even integer greater than $2$ is the sum of two primes


implies Goldbach's Weak Conjecture:

Every odd integer greater than $7$ is the sum of three odd primes.


Proof

Take any odd integer $n$ such that $n > 7$.

Then $m = n - 3$ is an even integer $n$ such that $m > 4$.


If the Goldbach Conjecture holds, then $m$ is the sum of two primes: $m = p_1 + p_2$.

If one of them were $2$, then $m - 2$ would have to be even, which if it is prime it can not be.

So if $m > 4$, both $p_1$ and $p_2$ must be odd.

So then we have that $n = p_1 + p_2 + 3$, that is, the sum of three odd primes.

So, if the Goldbach Conjecture holds, then so does Goldbach's Weak Conjecture.

$\blacksquare$