Modular Periodicity of the Euler Up/Down Numbers at Odd Prime Powers

Let E_n denote the number of alternating permutations of {1,...,n}, equivalently characterized by Σn≥0 E_n z^n/n! = sec z + tan z. For every q≥1, the sequence (E_n mod q)n≥0 is eventually periodic; let d(q) and s(q) denote its minimal eventual period and preperiod. For every odd prime p, Knuth and Buckholtz proved d(p)=lcm(p−1,4), together with d(p^r) | pʳ⁻¹d(p) and s(p^r)≤r, and Ramassamy conjectured that both bounds are attained for every r≥1. In this paper, we introduce an algebraic frequency expansion for the Euler numbers over S_r=(Z/p^rZ)[x]/(x^2+1). Using Hurwitz series, the Euler sequence is represented algebraically as a finite combination of formal exponential modes, in a manner reminiscent of Fourier analysis. Using this expansion, we prove d(p^r)=pʳ⁻¹d(p) for every odd prime p and every r≥1, thereby establishing Ramassamy's period conjecture. We also disprove the preperiod conjecture by proving s(5^5)≤4<5. Finally, we prove that 5^5 is the smallest odd prime power for which s(p^r)≠r, and based on our findings we conjecture s(p^r)≥r−2 for every odd prime p and every r≥2.