MathNet 2023-part-02 — 191 problems (EN–VI) Question 158

Let $u, v, w$ be positive integers, with $v$ and $w$ not exceeding $2023$, such that $u - v = 2w$ and $(u^2 + w^2)(v^2 + w^2)$ is divisible by $289$. How many sets of possible values of $(u, v, w)$ are there?