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I saw a number theory problem that began with "Fix $\pi \in Q - Q^2$; then $\pi^m \in Q^m - Q^{m+1}$." Here $Q$ is a prime ideal of a Dedekind domain, so this is obvious by unique prime factorization.

It seems "obvious," but I tried to write down a completely general proof and was a little baffled. Are there counterexamples for more general rings where $\pi \in Q - Q^2$ yet $\pi^m \in Q^{m+1}$, where $Q$ is a prime ideal or maximal ideal?

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