Prove that the following conjecture is equivalent to the strong Goldbach conjecture:
Every integer $n>3$ is halfway between $2$ primes.
I'm able to prove it, but i don't have much experience in writing proofs, witch is why i need help to find a proper way to explain it. I'd like to have a proof that is as "short and sweet" as the conjecture itself. The shorter the better!
What i have so far:
If $p$ and $q$ are a Goldbach's partition of an even integer $2n$, then:
$$ 2n=p+q $$
The midpoint between $p$ and $q$ is:
Therefore, if an even integer $2n$ can be written as the sum of $2$ primes, $n$ is halfway between those $2$ primes.