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

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.

13 pages

References in corpus (2)

• The 2-adic valuation of the coefficients of a polynomial
• The p-adic valuations of sequences counting alternating sign matrices

Cited by in corpus (3)

• Solving Quadratic and Cubic Diophantine Equations using 2-adic Valuation Trees
• 2-adic Valuations of Quadratic Sequences
• On the 2-adic valuation of (a^b-c^d)