Is there a proof which shows that for all primes p, n^p-n is divisible by p?

I've shown that for all positive integers n that n^5-5 is divisible by 5.

I've also found that this seems to be the case for p|(n^p-n), given that p is one of the first five primes. And that it is not true that p divides n^p-n when p is a composite positive integer.