Modular Periodicity of the Euler Up/Down Numbers at Odd Prime Powers: A Proof of the Period Conjecture and a Counterexample to the Preperiod Conjecture

For n ≥ 0, let Eₙ denote the number of alternating permutations of {1, …, n}, i.e. permutations π = (π₁, …, πₙ) satisfying π₁ < π₂ > π₃ < π₄ > ···. These are the Euler up/down numbers, whose exponential generating function is sec z + tan z. For an integer q ≥ 1, the sequence (Eₙ mod q) is ultimately periodic. We write s(q) for its preperiod and d(q) for its minimal eventual period. Knuth and Buckholtz proved that, for every odd prime p, d(p) = lcm(p − 1, 4), and that for every r ≥ 1,s(pʳ) ≤ r and d(pʳ) divides pʳ⁻¹d(p). Ramassamy conjectured that both bounds are attained. In this work, we prove the period conjectured(pʳ) = pʳ⁻¹d(p)for every odd prime p and every r ≥ 1. In contrast, we disprove the preperiod conjecture by proving that s(5⁵) ≤ 4, and hence s(5⁵) ≠ 5. A computational search further identifies 5⁵ as the smallest odd prime power for which s(pʳ) ≠ r.