Find all $n$ such that $n|1^n + 2^n + 3^n + \cdots + (n-1)^n$ where $n \in \mathbb{Z}^+$.

Question

Find all $n$ such that $$n|1^n + 2^n + 3^n + \cdots + (n-1)^n$$ where $n \in \mathbb{Z}^+$.

I don't know how to start. $n = 3, 5$ are simple solutions. Induction seems strange since the divisor is changing every time. How do I go about it?

Thanks.

Answer 1

All odd numbers $n$ are solutions to this. Take $n=2k+1$ and see what remainder the first and last terms give when divided by $n$ (it's $1^{2k+1}$ and $(-1)^{2k+1} = -1^{2k+1}$). Then see what remainder the 2nd and the 2nd last terms give (it's $2^{2k+1}$ and $(-2)^{2k+1}=-2^{2k+1}$) and so on.