MathNet 2025-part-01 — 187 problems (EN–VI) Question 3

Let $a$, $b$ and $k$ be positive integers. Suppose that for any integer $n \ge 2025$ there exists a positive integer $x_n \gt n$ such that $x_n \mid n^2 + a$ and $x_n + k \mid n^2 + b$. Prove that $k = b - a$.