Solution of $n!=p+1 $ with $p$ is prime number?

Question

One of my friend asked me to solve this equation $n!=p+1 $ with $p$ is prime number and n is positive integer , it's clear that for $p=2$ there is no solutions because : $n! < 3$ for $n=1$ , But what about $p >2$ ? Probably the solution of that equation w'd be satisfied with Mersann primes of the form $2^{p'}-1$ with $p'\neq 11$ , The reason i have got is $p=2^{p'}-1$ is a solution of $n!=p+1 $ because $n!$ never be a perfect square .

Answer 1

Examine some values $n!$ and try subtracting 1 from each of them: $$ 3! - 1 = 5, \;\mbox{a prime} $$ $$ 4! - 1 = 23, \;\mbox{a prime} $$ $$ 6! - 1 = 719, \;\mbox{a prime} $$ $$ 7! - 1 = 5039, \;\mbox{a prime}. $$