{
  "schemaVersion": "1.2",
  "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://evidencepress.org/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://evidencepress.org/releases/erdos-848-all-n/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/erdos-848-all-n/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/erdos-848-all-n.mp3",
  "imageUrl": "https://evidencepress.org/assets/og/erdos-848-all-n.png",
  "coverArtUrl": "https://evidencepress.org/assets/art/erdos-848-all-n.svg",
  "media": [
    {
      "type": "video",
      "url": "https://youtu.be/h2dUsd0F-Sk",
      "name": "Video explainer — Erdős 848: an exact answer for every N",
      "description": "Video explainer for this release on the Evidence Press YouTube channel."
    }
  ],
  "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."
  },
  "assurance": [
    {
      "dimension": "availability",
      "label": "Availability and archiving",
      "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?",
      "state": "passed",
      "evidenceUrl": "https://zenodo.org/records/21647629",
      "note": "Archived Zenodo deposit under a DOI."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "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."
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed"
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed"
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed"
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed"
    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
      "state": "not-assessed"
    },
    {
      "dimension": "dataEnvironmentReproducibility",
      "label": "Data and environment reproducibility",
      "question": "Are data and computational environment pinned well enough to rebuild?",
      "state": "not-assessed"
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": 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"
  },
  "corrections": [
    {
      "date": "2026-08-05",
      "scope": "presentation",
      "fixedIn": "Evidence Press 1.1.0",
      "summary": "From publication until 5 August 2026 this page displayed the main result as holding “for every $N$ from $\\\\lfloor (N+18)/25 \\\\rfloor$ upwards”. It should have read, and now reads, “for every $N$ from 1 upwards”.",
      "detail": "The cause was a defect in this site's Markdown renderer, which used plain integers as internal placeholders for mathematics and could therefore replace a literal number in prose with an unrelated formula. The error was introduced in rendering only: the archived deposit, the manuscript PDF, and this release's Markdown and JSON representations always carried the correct statement. No result, proof, certificate or item of evidence is affected, and the release's assurance state is unchanged."
    }
  ],
  "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)."
  ],
  "reviews": [],
  "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"
    }
  ]
}