{
  "schema_version": 1,
  "problem_number": "OPG-37413",
  "title": "A Computer-Assisted Proof of Alexa's Primality Criterion",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For every integer n>=8, put K=floor(n^(1/3)/2) and take r_i to be the least nonnegative residue of (2i+1)^(n-1) modulo n. We prove Alexa's corrected criterion: n is prime if and only if the ordinary integer sum of r_i for i=1,...,K equals K. Residue counting excludes cancellation by zero terms. A classical odd quadratic nonresidue bound excludes even inputs, and prime-product counting treats large prime squares. An integer-parameter version of the Forbes-Kayal-Mittal-Saha polynomial construction bounds an initial odd run in a proper finite-field subgroup of order g by 7sqrt(g)+50, forcing both factors of a remaining semiprime below 150000. Exact computations exclude all remaining cases: 1683064 odd primes in the square domain and 4571 base-3-liar semiprime pairs. C++ and Python implementations and witness certificates are supplied. This is a computer-assisted, AI-assisted, self-audited and unrefereed proof; no independent review, formal verification, practical primality-testing speedup or absolute priority certificate is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.NT"
  ],
  "keywords": [
    "OPG-37413",
    "primality criterion",
    "Alexa conjecture",
    "finite-field subgroup",
    "quadratic nonresidue",
    "computer-assisted proof"
  ],
  "manuscript_version_date": "2026-10-25",
  "publication_date": "2026-10-25",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-25",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/opg-37413/",
  "pdf_url": "https://eulersolve.org/papers/opg-37413/paper.pdf?v=3d6c3de58831",
  "doi": "10.5281/zenodo.23635918",
  "zenodo_record_url": "https://zenodo.org/records/23635918",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every integer n>=8, let K=floor(n^(1/3)/2) and r_i be the least nonnegative residue of (2i+1)^(n-1) modulo n. Then n is prime if and only if the ordinary integer sum of r_i, for i=1,...,K, equals K. This is the corrected 2013 Garden statement. This proves the corrected 14 June 2013 criterion with least nonnegative residues and an ordinary integer sum. The proof uses exact finite computations supplied in both C++ and Python, in addition to analytic arguments. Established source theorems and prior local reductions are credited. Absolute priority is not certified. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "3d6c3de58831310811f05899c5758a8518404580ded0bf0d4ba6a64eb1d67901"
    },
    "source.zip": {
      "sha256": "1c3a2f852701878eafaaa1bc9e89a7970ebab79679c8e42db7c2fe703df9d8d8"
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    "verification_report.md": {
      "sha256": "0d66f7634bb005139de399768b0d38f23e2bb40b71f109c2fbd903d23a482ea0"
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  },
  "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."
}