Periodicity in the \(p\)-adic valuation of a polynomial

by Luis A. Medina, Victor H. Moll and Eric Rowland

Journal of Number Theory , 180, 2017, 139-153.
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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