Find all pairs of positive integers $(m, n)$ such that $\sum_\limits{k=0}^{m} x^k \mid \sum_\limits{k=0}^{m} x^{kn}$

Question

I've found that $(m,1)$ works and $(1,n)$ works for all odd $n$. How can I identify all pairs $(m,n)$ that satisfy the expression?

Answer 1

This is my first response, so I apologize for my poor form!

It is easier to examine $x^{m+1}-1$ and $x^{n(m+1)}-1$ which we arrive at by multiplying the terms $\sum_{k=0}^mx^k$ and $\sum_{k=0}^mx^{kn}$ by $x-1$ and $x^n-1$, respectively. It is clear that $x^{m+1}-1|x^{n(m+1)}-1$. The question becomes this: what are the common factors of $x^{m+1}-1$ and $x^n-1$? This is much easier to handle, and the answer is almost immediate. As long as $gcd(m+1,n)=1$, the only common factor of $x^{m+1}-1$ and $x^n-1$ is $x-1$. It follows that if $gcd(m+1,n)=1$, then $\sum_{k=0}^mx^k|\sum_{k=0}^mx^{kn}$.