MathNet 2026 — Part 08 Question 80

Let $x$, $y$, and $z$ be positive integers satisfying $x y = z^{2} + 1$. Prove that there are integers $a$, $b$, $c$, and $d$ such that $x = a^{2} + b^{2}$, $y = c^{2} + d^{2}$, and $z = a c + b d$.