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

Question

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.

? · Accepted Answer

Factor n^p  - n into n(n^[ p-1 ] - 1).

Then, according to Fermat's "little" theorem,

p | n^(p-1) - 1

DWRead · Answer

n^5-5 is not divisible by 5 for n = 1.

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

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