TITLE: A condition for being a prime: $\;\forall m,n\in\mathbb Z^+\!:\,p=m+n\implies \gcd(m,n)=1$
QUESTION [14 upvotes]: If $\;p=m+n$ where $p\in\mathbb P$, then $m,n$ are coprime, of course. But what about the converse?
Conjecture:

$p$ is prime if $\;\forall m,n\in\mathbb Z^+\!:\,p=m+n\implies \gcd(m,n)=1$

Tested (and verified) for all $p<100000$.

REPLY [4 votes]: for p = 1 obviously wrong
(for all positive integers m, n with m+n=p (of course there are no ones, doesn't matter) there is gcd(m,n)=1, but 1=p is not prime)