MathNet 2024-part-04 — 168 problems (EN–VI) Question 64

Let $n$ be a positive integer and let $a$ and $b$ be positive integers congruent to $1$ modulo $4$. Prove that there exists a positive integer $k$ such that at least one of the numbers $a^k - b$ and $b^k - a$ is divisible by $2^n$.