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Beautiful number theory problem

Source: BMO 2023 Problem 3

May 10, 2023
number theoryBMO 2023gauss lemma

Problem Statement

For each positive integer nn, denote by ω(n)\omega(n) the number of distinct prime divisors of nn (for example, ω(1)=0\omega(1)=0 and ω(12)=2\omega(12)=2). Find all polynomials P(x)P(x) with integer coefficients, such that whenever nn is a positive integer satisfying ω(n)>20232023\omega(n)>2023^{2023}, then P(n)P(n) is also a positive integer with ω(n)≥ω(P(n)).\omega(n)\ge\omega(P(n)).
Greece (Minos Margaritis - Iasonas Prodromidis)
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