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Using Fermat's Theorem Prove if $p$ is prime, prove $1^p + 2^p + 3^p +...+(p-1)^p \equiv 0 \bmod{p}$

The two definitions of Fermat's Little Theorem is $a^p \equiv a \bmod{p}$ and $a^{p-1} \equiv 1 \bmod{p}$ but I don't know how to use this solve the problem

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Hint: Use Fermat's theorem for each $k$ ($1\le k\le p-1$) and add all them up. Then use Gauss's formula for the sum $1+2+\cdots +(p-1)$.

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Since $1^p \equiv 1$, and $2^p \equiv 2$, $\ldots$ $(p-1)^p \equiv p-1$, we have that

$$1^p + 2^p + \ldots (p-1)^p \equiv 1 + 2 + \ldots + p-1$$

However, we know this sum: $\sum_{i = 1}^{p-1} i = \frac{p (p-1)}{2}$, which for odd primes would be a multiple of $p$, and we are done by reducing modulo $p$, then we only must check the case for $2$, which doesn't hold, and it's awkward...

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$1^p + ... + (p-1)^p \equiv 1 + 2 + ... (p-1) \equiv (1 + (p-1)) + (2 + (p-2)) + ... +(\frac{p-1}{2} + \frac{p+1}{2}) \equiv p + ... + p \equiv 0 (\mod p) $

Edit: Assuming p is an odd prime, of course

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I understand the 1+(p-1)+2+(p-2)+.... part but how did you come up with (p-1/2 + p+1/2)? – Cindy Dec 10 '12 at 4:54
Because $p$ is odd, that'll be the last two pairs in the sum. Write out some of these sums for various primes and you'll see where it comes from. – anonymous Dec 10 '12 at 16:39

Before you wonder if you know how to use Fermat's theorem to solve the problem, first wonder if you are able to use Fermat's theorem at all.

There are a lot of things raised to the $p$-th power, and one form of Fermat's theorem tells you something about things raised to the $p$-th power. So you see what Fermat's theorem tells you about them.

You don't do this because you know it will solve the problem: you do this because maybe it will tell you something useful.

As it turns out, the thing it tells you is that your problem is equivalent to a problem you already know how to solve.

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There is no need of Fermat's theorem. If $p$ is odd, then $p$ divides $$ k^p +(p-k)^p = (k+p-k)\left(k^{p-1}-\ldots+(p-k)^{p-1}\right)=p A, $$ hence by collecting the terms of your sum in couples (just like Gauss did when he was five years old in order to prove that $1+2+\ldots+n = \frac{(n+1)n}{2}$) you get that the sum is a multiple of $p$.

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