MathNet 2026 — Part 10 Question 23

Let $P(x)$ be a polynomial with integer coefficients. Let $a_{0}=0$ and for $i \geq 0$ define $a_{i+1}=P\left(a_{i}\right)$. Prove that $\operatorname{gcd}\left(a_{m}, a_{n}\right)=a_{\gcd(m, n)}$ for any integers $m, n \geq 1$.