A Divisor problem for polynomials

TL;DR

This paper characterizes monic integer polynomials that divide their values at p^p for large primes p, providing necessary and sufficient conditions for such divisibility to hold for all sufficiently large primes.

Contribution

It offers a complete characterization of polynomials with the divisibility property and establishes necessary and sufficient conditions for all primes.

Findings

Characterization of polynomials with the divisibility property

Necessary conditions for the divisibility property

Sufficient conditions for the divisibility property

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 $p$.