Lower bound for the preperiod of Euler up/down numbers modulo odd prime powers

Prove that for every odd prime p and every integer r≥2, the preperiod s(p^r) of the Euler up/down numbers modulo p^r satisfies s(p^r)≥r−2.

Background

For an odd prime p, let s(p^r) denote the minimal preperiod of the sequence of Euler up/down numbers (E_n modulo p^r). The paper disproves the previously conjectured equality s(p^r)=r by establishing s(5⁵⁾⁼⁴ and identifies 5⁵ as the smallest counterexample. The authors then derive a divisibility criterion that characterizes the possible reductions of the preperiod through conditions of the form p^{j∣E_{r−j}.}

Computational searches found many cases with s(p^r)=r−1 and one case with s(43^{980)=978=r−2,} but no examples with preperiod at most r−3 among odd prime powers p^{r≤10^100000.} These observations motivate the conjectured universal lower bound s(p^r)≥r−2.

References

This naturally led us to ask whether there are cases in which the preperiod is r-2. A search designed specifically to detect such cases found no example up to 10^{1500}. Extending the range to 10^{2000}, however, produced the example s(43^{980})=978. We then asked whether there are cases with preperiod r-3. Rather surprisingly, an exhaustive search found no such example among odd prime powers p^{r\le10^{100000}.} Therefore, these computations suggest that the phenomenon may be universal, leading us to the following conjecture. For every odd prime p and every r\ge2, s(p^r)\ge r-2.

— Modular periodicity of the Euler up/down numbers at odd prime powers (2608.27058 - Güleç, 27 Aug 2026) in Section 5, Computational remarks, Conjecture 5.5 (following the computational discussion)

Lower bound for the preperiod of Euler up/down numbers modulo odd prime powers

Background

References

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