  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.
