Here is a small cute lemma, that I have encountered while solving a problem. Hope you will enjoy it.
Let $p$ be an odd prime. Define $S_k$ via $$ S_k \triangleq \sum_{i=1}^{p-1}i^k. $$ Show that $p \mid S_k$ for all $1 \leq k < p-1$.
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3I'm sorry, but what is the equal sign with the birthday hat? I've never seen it before. Also, Happy Birthday equal sign! – Chickenmancer May 04 '17 at 14:14
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What do you mean by $ \triangleq$? – Shaun May 04 '17 at 14:14
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1I think it means define $S_k$ to be that sum. – Zain Patel May 04 '17 at 14:15
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Does it mean "is defined as being equal to"? – Shaun May 04 '17 at 14:16
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You're better off telling us what you've done to show that yourself. You might get close votes otherwise. – Shaun May 04 '17 at 14:18
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Here is a different proof from the links shown. Hope this is correct.
If $k$ is odd, then $i^k + (p-i)^k = 0 \mod p$
Consider $k$ even:
$1^k, 2^k, ..., (p-1)^k$ are all quadratic residues$\mod p$ with the values being equal WLOG to $1, g, g^2, ..., g^{(p-3)/2}$ and whose sum is $g^{(p-1)/2} - 1$ which is $0 \mod p$ per Fermat and where $g$ is a quadratic residue.
sku
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