A result of Monsky states that the Hilbert–Kunz function of a one-dimensional local ring of prime characteristic has a term
ϕ
\phi
that is eventually periodic. For example, in the case of a power series ring in one variable over a prime-characteristic field,
ϕ
\phi
is the zero function and is therefore immediately periodic with period 1. In additional examples produced by Kunz [Amer. J. Math. 91 (1969), pp. 772–784] and Monsky [Math. Ann. 263 (1983), pp. 43–49],
ϕ
\phi
is immediately periodic with period 2. We show that, for every positive integer
π
\pi
, there exists a ring for which
ϕ
\phi
is immediately periodic with period
π
\pi
.