In this (btw, nice) answer to Twin primes of form $2^n+3$ and $2^n+5$, it was said that: If $n\equiv 2\pmod 3$, then $7\mid 2^n+3$? I'm not familiar with these kind of calculations, so I'd like ...
Theorem: $(n - 1)! + 1 \equiv_n 0$ if and only if $n$ is prime. To prove that if $n$ is not prime this is not true is trivial, so I'm just interested in proving that this is true for all p: ...