Prove P is prime

[Zzzz]

If $P$ divides $(P-1)!+1$, prove that $P$ is a prime.

[Hasib]

nice problem to face.

[Tahmid Hasan]

i solved it once so i'll not post the soln,but here' a hint.
what are the properties of prime ad odd numbers?

[photon]

suppose P isn't prime.P has some divisors.these divisors must be be lesser than P-1.
then P divides (P-1)!.but P divides (P-1)!+1.
so P cannot be non-prime.
P is prime.

[Hasib]

its reverse once is also true. If p is a prime, then p divides (p-1)!+1 the wilson theorem

[Moon]

Certainly this is wilson's Theorem; however, what Zzzz wanted to indicate is that the reverse of this theorem is also true.

[Masum]

If $P$ divides $(P-1)!+1$, prove that $P$ is a prime.

Well, if $n|(n-1)!+1$ with $n=ab,a,b>1$ then $a<n-1$ and so $a|(n-1)!$ and also $a|n|(n-1)!+1$ implying that $a|1$, a clear contradiction!

[mahathir]

well,we know,if $p$ is not a prime,then it will have a prime divisor between $1$ and \[\sqrt{p}\] so, it will share a common factor with $(p-1)$ but not with $1$. so,$p$ cannot be a composite number.Hence,since it has to be a number(the question states that it does divide) so,it will be a prime number.
Also,check wilson's theorem,In mathematics, Wilson's theorem states that a natural number n > 1 is a prime number if and only if \[(n-1)!\equiv -1(mod n)\]