Problem

Source: Indonesian Stage 1 TST for IMO 2022, Test 2 (Number Theory), Canadian MO Qualification Repechage 2018 No 7

Tags: polynomial, number theory, prime, Product, infinite



Let $n$ be a natural number, with the prime factorisation \[ n = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r} \]where $p_1, \ldots, p_r$ are distinct primes, and $e_i$ is a natural number. Define \[ rad(n) = p_1p_2 \cdots p_r \]to be the product of all distinct prime factors of $n$. Determine all polynomials $P(x)$ with rational coefficients such that there exists infinitely many naturals $n$ satisfying $P(n) = rad(n)$.