Prove that n!-1 is a composite number when n>3 and n+2 is a prime.
If n+2 is prime, then by Wilson's theorem
(n+1)! = -1 (mod (n+2))
=> n! = 1 (mod (n+2))
=> n!-1 =0 (mod (n+2))
If (n+2) is prime, then n!-1 is divisible by (n+2)
making n!-1 a composite number.
Posted by Praneeth
on 2007-09-27 06:01:18