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.
$\blacksquare$