For any number n, if the square root of (n!)+1 is an integer, it is also prime. (This is interesting, but rather useless in practice)
For any number a, 1<a<=n, n! mod a == 0; therefore, n!+1 mod a == 1. n!+1 is prime. Prime numbers don't have integral square roots.
You're getting things missed up with the classic proof that there is no largest prime number. This doesn't hold in general. Try a=5. 5!=5*4*3*2*1=120. 120+1=121. 121=11*11. The classic proof goes: Is there a largest prime number? If there is then collect all primes, p1...pn and multiply them together p=p1*p2*...*pn. p+1 is not divisible by p1...pn. Therefore p+1 is a prime. Therefore there is no largest prime number.
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