Prime number Determinants

Question

I have come up this final result after deriving some prime-related equations. Let $m,s,P$, be an integers, and $P = 6k+1, k>0$. Let $m^2 = 9s^2 + P$, then $P$ is prime iff $\forall s, -k<s<k$, there is no integer solution for $m$. Alternatively, if there exists at least one integer $s, -k<s<k$, so that $m$ is an integer, then $P$ is nonprime. Can someone verify this result if I am correct or not?