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arXiv · 1505.02302

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

Luis A. Medina·Victor H. Moll·Eric Rowland
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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.

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Luis A. Medina, Victor H. Moll, Eric Rowland. 2015-05-09. Periodicity in the $p$-adic valuation of a polynomial. https://doi.org/10.1016/j.jnt.2017.03.006

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