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A result of Monsky states that the Hilbert-Kunz function of a one-dimensionallocal ring of prime characteristic has a term $\phi$ that is eventuallyperiodic. For example, in the case of a power series ring in one variable overa prime-characteristic field, $\phi$ is the zero function and is thereforeimmediately periodic with period 1. In additional examples produced by Kunz andMonsky, $\phi$ is immediately periodic with period 2. We show that, for everypositive integer $\pi$, there exists a ring for which $\phi$ is immediatelyperiodic with period $\pi$.

A Hilbert-Kunz function with a periodic term that has a given period