After seeing and doing a bunch of proofs like "For all $a$ in the natural numbers, then if $7$ does not divide $a$, then $7$ divides $a^3+1$ or $a^3-1$," I conjectured the following, but got stuck in proving it. I'd like to know if it has an actual name, or whether this is just something rather trivial and pointless.
Given a prime $p$ of the form $2n+1$, for all $a$ in the natural numbers if $p$ does not divide $a$, then $p$ divides $a^n+1$ or $a^n-1$.
*Assume non-trivial $a$
Thanks very much.