I do not know how to format equations on this website (this is my first question), so please bear with me and point me in the direction on how to do this in the future.
I have been working on a proof, and I have managed to bring it to the point where if I can prove that $8p! + (2p-1)^2 = b^2$, for $p\geq3$ has no solution such that $b$ and $p$ are integers, I will have a complete proof. It is the same as saying $8p! + (2p-1)^2$ can not equal a perfect square for any integer $p$ greater than $3$.
I have tried using the fact that any perfect square is equal to $1$ or $0$ mod $3$, but the expression above is also always equal to $1$ or $0$ mod $3$.
I have no idea how to do this, and the factorial is giving me issues. I was wondering if someone could point me in the right direction or give me some known facts that can be used to prove something like this. Much appreciated.