Odd-prime-power periodicity conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Define the Euler up/down numbers $E_n$ for integers $n\ge0$ by \[ \sum_{n\ge0} E_n\frac{z^n}{n!}=\sec z+\tan z. \] For $q\in\mathbb N$ this sequence modulo $q$ is eventually periodic. Its minimal preperiod is \[ s(q)=\min\{s\in\mathbb Z_{\ge0}:\exists d\in\mathbb N\ \forall n\ge s,\ E_{n+d}\equiv E_n\pmod q\}, \] and its minimal eventual period is \[ d(q)=\min\{d\in\mathbb N:\forall n\ge s(q),\ E_{n+d}\equiv E_n\pmod q\}. \] The prime-level periods are $d(p)=p-1$ for $p\equiv1\pmod4$ and $d(p)=2(p-1)$ for $p\equiv3\pmod4$.

Historical context. These are the two parts of Ramassamy's original conjecture. A 2026 paper reports a proof of the period formula and the counterexample $s(5^5)=4<5$ to the preperiod formula. [S3]

Statement recorded in the supplied source.

For every odd prime $p$, does $s(p^k)=k$ hold for every integer $k\ge1$, and does $d(p^k)=p^{k-1}d(p)$ hold for every integer $k\ge2$? [S2]

Short English statement

Determine the exact starting point and period of the Euler-number sequence modulo every odd prime power; the original two predictions are recorded with their later source context.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9100001 (AMR-090-0001). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Sanjay Ramassamy, Modular periodicity of the Euler numbers and a sequence by Arnold, Arnold Mathematical Journal (2018) §§1--2, Conjecture 1, pp. 1--3. https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/18-79/index.html.
• S3 Berke Güleç, Modular periodicity of the Euler up/down numbers at odd prime powers, arXiv:2608.27058v2 (2026) Theorems 5.9 and 6.4. https://arxiv.org/pdf/2608.27058.