If $p$ is a prime number and a is a positive integer, show that $a^{(p-1)!+1} \equiv a\pmod{p}$

If $p$ is a prime number and a is a positive integer, show that $$a^{(p-1)!+1} \equiv a\pmod{p}$$ I think Wilson's theorem might related to this problem because $(p-1)!+1$ is similar to $(p-1)! \equiv -1 \pmod{p}$.

Can anyone can give a hint or some part of proof for this problem?

• Wilson's theorem is not relevant in any obvious way, since you care about the exponent mod $p-1$, not mod $p$. – Eric Wofsey Sep 21 '16 at 8:28
• If $p$ does not divide $a$, then this follows directly from Fermat's little theorem (or the more general Euler's theorem), since as you've mentioned, you can rewrite $(p-1)!+1$ as $kp$ for some $k\in\mathbb{N}$. – barak manos Sep 21 '16 at 8:48

Hint: Use Fermat's little theorem. In particular, if $a$ is not divisible by $p$, what can you say about $a^{(p-1)!}$?

Here is a funny variation :

The first case $p\mid a$ is trivial (look at @fleabood argument).

For the second case $p\not \mid a$ we will prove this statement :

Let $k,l$ two integers and $u$ an integer such that $p\not \mid u$. If $k\equiv l \ [p-1]$ then $u^k\equiv u^l\ [p].$

To prove this : we traduce the first congruence. It means that there exists $x\in \mathbb{Z}$ such that $(p-1)x+l=k$.

Then we can say that $u^k=u^{(p-1)x+l}=(u^{(p-1)})^{x}u^l$. By applying Fermat little theorem we have finally that : $u^{k}\equiv u^l \ [p]$.

Now we can apply the statement to : $a=u,k=(p-1)!+1$ and $l=1$ because $(p-1)!+1\equiv 1 \ [p-1]$ and it gives the final answer.

Remark : If we wanted to use the powerful Wilson's theorem we will have $(p-1)!+1\equiv 0 \ [p]$. But the previous statement is not sufficient. Indeed, in the last congruence, we will have $u^x$ which is totally unknown. If we suppose $p=k-l$, we will obtain $u^k\equiv u^{l+1} \ [p]$.

This is a very easy result gussied up to look hard.

Case 1: $p|a$

Then $a \equiv 0 \mod p$ and $a^N \equiv a \equiv 0 \mod p$ for any $N$.

Case 2: $p \not \mid a$ then as $p$ is prime $\gcd(a,p) = 1$

and $a^{p-1} \equiv 1 \mod p$ via Fermats Little Thereom.

So $a^{N(p-1) + 1} \equiv (a^{p-1})^Na \equiv a \mod p$ of any $N$ (including, but not limited to, $N = (p-2)!$).