MathNet 2007-part-02 — 187 problems (EN–VI) Question 42

Let $b$, $n \gt 1$ be integers. Suppose that for each $k \gt 1$ there exists an integer $a_{k}$ such that $b - a_{k}^{n}$ is divisible by $k$. Prove that $b = A^{n}$ for some integer $A$.