Periodicity in the $p$-adic valuation of a polynomial

TL;DR

This paper investigates the behavior of the p-adic valuation sequence of polynomial values, showing it is either periodic or unbounded, with the period length explicitly determined when periodic.

Contribution

It characterizes the conditions under which the p-adic valuation sequence of a polynomial is periodic and provides a method to determine the period length.

Findings

The valuation sequence is either periodic or unbounded.

Period length is explicitly determined when the sequence is periodic.

No roots in p-adic integers imply unbounded valuations.

Abstract

For a prime $p$ and an integer $x$, the $p$-adic valuation of $x$ is denoted by $\nu_{p}(x)$. For a polynomial $Q$ with integer coefficients, the sequence of valuations $\nu_{p}(Q(n))$ is shown to be either periodic or unbounded. The first case corresponds to the situation where $Q$ has no roots in the ring of $p$-adic integers. In the periodic situation, the period length is determined.