{
  "schemaVersion": "1.1",
  "slug": "erdos-848-all-n",
  "title": "Erdős problem 848: an exact answer for every N",
  "shortTitle": "Erdős 848: exact answer for all N",
  "url": "https://evidence-press.pages.dev/releases/erdos-848-all-n/",
  "oneLine": "A certificate-backed determination that the answer to an Erdős–Sárközy extremal problem is exactly ⌊(N+18)/25⌋ for every N — closing the gap between known asymptotic results and small cases.",
  "abstract": "Erdős and Sárközy asked: how large can a set A of integers from 1 to N be if the product of any two members (including a member with itself), plus one, is never squarefree? Recent work resolved the question for all sufficiently large N. This release establishes the exact answer f(N) = ⌊(N+18)/25⌋ for every positive integer N, stitching together exact colouring certificates for small N, structural decompositions and exact-rational envelope arguments for intermediate ranges, and a pinned explicit-threshold analytic theorem for N beyond 2.64 × 10^17.",
  "datePublished": "2026-07-28",
  "dateModified": "2026-07-28",
  "version": "0.1-candidate",
  "doi": "10.5281/zenodo.21647629",
  "doiUrl": "https://doi.org/10.5281/zenodo.21647629",
  "conceptDoi": "10.5281/zenodo.21647628",
  "pdfUrl": "https://github.com/ipitchford/erdos-848-all-n/releases/download/v0.1-candidate/paper.pdf",
  "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/erdos-848-all-n/main/paper.pdf",
  "zenodoUrl": "https://zenodo.org/records/21647629",
  "repoUrl": "https://github.com/ipitchford/erdos-848-all-n",
  "releaseUrl": "https://github.com/ipitchford/erdos-848-all-n/releases/tag/v0.1-candidate",
  "markdownUrl": "https://evidence-press.pages.dev/releases/erdos-848-all-n/index.md",
  "bibtexUrl": "https://evidence-press.pages.dev/releases/erdos-848-all-n/cite.bib",
  "audioUrl": "https://evidence-press.pages.dev/assets/audio/erdos-848-all-n.mp3",
  "imageUrl": "https://evidence-press.pages.dev/assets/og/erdos-848-all-n.png",
  "coverArtUrl": "https://evidence-press.pages.dev/assets/art/erdos-848-all-n.svg",
  "media": [],
  "authors": [
    "OpenAI Codex"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Unrefereed candidate result. The release itself states it is not independent external reproduction, external human peer review, or end-to-end formal verification; correctness rests on local certificate replay under documented compiler, runtime, and hardware assumptions, and on an external third-party explicit-threshold theorem (a pinned PDF source). The catalogue at erdosproblems.com records the resolution for sufficiently large N but does not (as of this release) acknowledge an all-N determination."
  },
  "provenance": {
    "aiGenerated": true,
    "generatedBy": [
      "OpenAI Codex"
    ],
    "humanRole": "problem selection, mediation, and publication management",
    "disclosure": "The mathematics/research in this release was generated by AI systems as credited; see the Zenodo record for full attribution."
  },
  "problem": {
    "name": "Erdős problem 848 (Erdős–Sárközy, nonsquarefree ab+1)",
    "url": "https://www.erdosproblems.com/848"
  },
  "keywords": [
    "Erdős problem 848",
    "squarefree numbers",
    "extremal number theory",
    "combinatorial number theory",
    "computer-assisted proof",
    "certificate replay",
    "exact-rational certificates",
    "AI-generated mathematics",
    "reproducible research"
  ],
  "keyResults": [
    "Claim: f(N) = ⌊(N+18)/25⌋ for every positive integer N, where f(N) is the largest size of A ⊆ {1,…,N} with ab+1 nonsquarefree for all a, b ∈ A (including a = b).",
    "1 ≤ N ≤ 100,000,006: exact finite colouring certificates with compact endpoint induction.",
    "10^8 ≤ N ≤ 10^9: exhaustive structural decomposition with exact lower-range validation.",
    "10^9 ≤ N ≤ 10^12: exact-rational short-shift envelopes.",
    "10^12 ≤ N ≤ 2.64 × 10^17: exact-rational rank envelopes.",
    "N ≥ 2.64 × 10^17: an explicit analytic threshold theorem (external, pinned source)."
  ],
  "evidencePackage": "A 75.6 MB archived deposit containing LaTeX/PDF documentation, a Python replay framework with semantic mutations and sanitizer controls, a C++ verifier for the range up to 100,000,006 with a compressed colouring-delta binary, SHA-256 manifests for all principal components, and machine-readable claim indexes (AI_INDEX.json/md, STATUS, ASSURANCE, PROVENANCE).",
  "openProblems": [
    "Independently replay the certificates on different hardware and toolchains — the C++ verifier for N ≤ 10^8 with its compressed colouring-delta binary is the most accessible entry point.",
    "Independently verify the exact-rational envelope arguments covering 10^9 ≤ N ≤ 2.64 × 10^17, which are the least conventional part of the architecture.",
    "Confirm the hand-off: check that the pinned explicit-threshold theorem genuinely covers all N ≥ 2.64 × 10^17 under the same normalisation of f(N).",
    "Formalise the endpoint-induction scheme in a proof assistant.",
    "Ask whether the same five-regime architecture (certificates, structural decomposition, envelopes, analytic threshold) transfers to neighbouring extremal problems such as Erdős problem 844."
  ],
  "relatedWorks": [
    {
      "citation": "Erdős problem 848 (Erdős & Sárközy): maximum size of A ⊆ {1,…,N} with ab+1 never squarefree — the problem page also presents van Doorn's upper bound |A| ≤ (0.108…+o(1))N.",
      "url": "https://www.erdosproblems.com/848"
    },
    {
      "citation": "Sawhney, M. Resolution for all sufficiently large N, with the structural statement that near-extremal sets lie in {n ≡ 7 (mod 25)} or {n ≡ 18 (mod 25)}.",
      "url": "https://www.math.columbia.edu/~msawhney/Problem_848.pdf"
    },
    {
      "citation": "Sothanaphan, N. An explicit threshold in Erdős Problem #848 — the pinned analytic input for N ≥ 2.64 × 10^17.",
      "url": "https://drive.google.com/file/d/1ujhm4_WYpgRV_rd1rJXIfHyvx16COEKe/view"
    }
  ]
}