{
  "schema_version": 1,
  "problem_number": "AMR-090-0002-0003",
  "title": "Euler Numbers Modulo Powers of Two and Arnold's Sequence: Proof of Two Conjectures of Ramassamy",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "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.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "keywords": [
    "AMR-090-0002-0003",
    "AMR-090-0002",
    "AMR-090-0003",
    "Euler numbers",
    "Entringer numbers",
    "2-adic valuation",
    "Arnold's sequence",
    "modular periodicity",
    "math.NT",
    "math.CO"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.1",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.1",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-090-0002-0003/",
  "pdf_url": "https://eulersolve.org/papers/amr-090-0002-0003/paper.pdf?v=e847ea4e3e26",
  "doi": "10.5281/zenodo.23049960",
  "zenodo_record_url": "https://zenodo.org/records/23049960",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Scope: This paper proves Ramassamy's Conjectures 2 and 3, corresponding to corpus records AMR-090-0002 and AMR-090-0003. The period formula is credited to classical congruences; no novelty claim is made for it. Conjecture 1 is not included. Arnold's 1991 and 2004 original texts were not fully inspected, so no absolute priority claim is made. Self-audited, AI-assisted and unrefereed; no independent peer review is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "e847ea4e3e26e6a7175b8e49e861e49158375155ede1937d4fdb9b7b470ac990",
      "bytes": 144705
    },
    "source.zip": {
      "sha256": "b0b565d0b62884bdf420a0de774bf30be8761e947cbed8d68c5a8d722489c41e",
      "bytes": 85399
    },
    "verification_report.md": {
      "sha256": "12484d0f375e60294d9488cec2947b1633d44c2b78fea5557e1f3f49a996e6f8",
      "bytes": 19105
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text.",
  "concept_doi": "10.5281/zenodo.23049669",
  "concept_url": "https://doi.org/10.5281/zenodo.23049669",
  "revision_published_at": "2026-09-30T01:46:06.384837+00:00",
  "version_history": [
    {
      "version": "1.0",
      "doi": "10.5281/zenodo.23049670",
      "zenodo_record_url": "https://zenodo.org/records/23049670",
      "files": {
        "paper.pdf": {
          "sha256": "46ca625cc75191b79a325258e668c588a5b9d93a8a38fca9e6c34c49e28ded83"
        },
        "source.zip": {
          "sha256": "cca689d77fd9730483b681797b3bd90cc3dc95738df92c15c98df5d642b8f194"
        },
        "verification_report.md": {
          "sha256": "ba294e839197d4cb9c7971cbea3796954dfdeee327e76ea2e29cf7e0ad3ca72c"
        }
      },
      "public_archive_preserved": true
    }
  ],
  "revision_note": "Arnold's observation is attributed as reported by Ramassamy; the classical Bernoulli-number expansion of tan x and DOI references are supplied; the OEIS remark and verification record are made precise.",
  "review_disclosure": "Internal AI-assisted checks only; unrefereed preprint, no independent human peer review claimed."
}