A Divisor problem for polynomials

Abstract

We characterize all monic polynomials $f(x)\in \mathbb{Z}[x]$ that have the property that \[f(p) \mid f(p^{p}),~\text{for all sufficiently large primes }p \geq N(f). \] We also give necessary conditions and a sufficient condition for monic polynomials $f(x)\in \mathbb{Z}[x]$ to satisfy $f(p)\mid f(p^p)$ for all primes $p$.

Acta Arithmetica 200 (2021), 111-118