MathNet 2022-part-01 — 197 problems (EN–VI) Question 111

Let $x \gt y \gt 2022$ be natural numbers such that $xy + x + y$ is a perfect square. Is it possible for each natural number $z$ in the interval $[x + 3y + 1, 3x + y + 1]$ the numbers $x + y + z$ and $x^2 + xy + y^2$ to not be coprimes?