MathNet 2017-part-05 — 176 problems (EN–VI) Question 173

Let $m, n, k$ and $l$ be positive integers with $n \neq 1$ such that $n^{k} + m n^{l} + 1$ divides $n^{k+l} - 1$. Prove that either $m = 1$ and $l = 2k$; or $l \mid k$ and $m = \frac{n^{k-l} - 1}{n^{l} - 1}$.