Euler Numbers Modulo Powers of Two and Arnold's Sequence: Proof of Two Conjectures of Ramassamy

Let E_n be the Euler up/down numbers, Σ_n E_n x^n/n! = sec x + tan x, and let e_{n,i} be the Entringer numbers, which form the Seidel–Entringer–Arnold triangle. Arnold observed, without proof, that the least 2-adic valuation m_i of the entries on the i-th diagonal of this triangle is weakly increasing in i, and he introduced the sequence u_k = max{i : m_i < k}. In 2017 Ramassamy conjectured that the sequence (E_n mod 2^k)_{n≥0} is periodic from the index u_k on but not from any earlier index, that its least period is 2^k for k ≠ 2 and 2 for k = 2, and that (u_k) is the f-transform of (2,4,4,4) for an explicit doubling map f. We prove these two conjectures. With h(j) = 2j − 2 − v_2(j), the 2-adic valuation of the tangent number E_{2j−1}, we show that m_i = min_{j≥⌈i/2⌉} h(j) for every i, which proves Arnold's observation, and that u_k = 2 max{j : h(j) < k}. The statements about the period, and the value 2 max{j : h(j) < k} of the preperiod, follow quickly from Stern's classical congruence for the Euler numbers and from the valuation of the tangent numbers. The new ingredients are the determination of m_i and u_k and the proof of the f-transform identity, which rests on the self-similarity h(2^{a−1} + r) = h(r) + 2^a for 1 ≤ r < 2^{a−1}. The proofs are elementary, and computations serve only as consistency checks. This is an unrefereed note. Version 1.1 (30 September 2026) revises version 1.0 after an independent referee round. Main changes: Arnold's observation is now attributed as reported by Ramassamy; the list of classical tools used in the proofs is completed (the Bernoulli-number expansion of tan x); DOIs are added; the remark on OEIS A108039 is made precise; and the verification record is corrected. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: AMR-090-0002, AMR-090-0003 (Ramassamy, Arnold Math. J. 3 (2017), Conjectures 2 and 3).