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 for all $p<100000$.
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It is true. Suppose $p\geqslant 2$ is not prime. Then we can write $p=xy$ with $x,y\geqslant 2$. Then we find $p=m+n$, with $m=x$ and $n=x(y-1)$. Those are obviously not coprime. |
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