Let $\ n\ $ be a positive integer and $\ p\ $ a prime number such that $$\ p^2\mid (2n)! + n! + 1$$ The only pairs $\ (n,p)\ $ I found so far are
$(1,2)$ , $(2,3)$ , $(10,11)$ , $(106,107)$ , $(4930,4931)$
In each pair we have $\ n=p-1\ $.
Can we prove that this is always the case ? Are there infinite many primes $\ p\ $ such that $$p^2\mid (2p-2)!+(p-1)!+1$$ ?