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Is this statement valid?

"If $k$ is a positive odd number, then $1^k+2^k+· · ·+n^k$ is divisible by $n+1$"

I don't think so. Did I missed anything subtle? (given who wrote the book)

Source: Solving Mathematical Problems by Terence Tao

Relevant.

Tian Vlasic
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    I mean, did you try the case $k=1$? – lulu Aug 22 '22 at 18:42
  • yes I did that and it didn't work. Are you confirming that it is invalid?

    If it was a misprint, what might be the intended statement?

     – nextQuestion Aug 22 '22 at 18:46
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    It's not even close to valid. The parity is wrong. Take $n$ odd of the form $4m+1$. Then the left hand is odd (for any $k$) but $n+1$ is even. – lulu Aug 22 '22 at 18:55
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    I suggest: try to prove that the parity problem is the only obstruction. That is to say, there are no counterexamples other than the (infinite) family I described. – lulu Aug 22 '22 at 19:08
  • You might want to look at this, which is the closest related thing I can think of: https://arxiv.org/abs/math/9207222. You always get a polynomial of the case $k=1$, that is $\frac{n(n+1)}{2}$. – Thomas Anton Aug 22 '22 at 20:59

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