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Polynomial on primes

Source: IMEO 2019, Problem 4

October 14, 2019
polynomialprime numbersalgebraalgebra proposed

Problem Statement

Call a two-element subset of N\mathbb{N} cute if it contains exactly one prime number and one composite number. Determine all polynomials fāˆˆZ[x]f \in \mathbb{Z}[x] such that for every cute subset {p,q} \{ p,q \}, the subset {f(p)+q,f(q)+p} \{ f(p) + q, f(q) + p \} is cute as well.
Proposed by Valentio Iverson (Indonesia)
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