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.
