{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-problem-126-prime-divisors-of-pairwise-sums",
  "title": "Erdős Problem #126: prime divisors of pairwise sums",
  "canonical_statement": "Let $f(n)$ be maximal such that for every $n$-element set $A\\subseteq\\mathbb N$,\n$$\n\\prod_{\\substack{a,b\\in A\\\\a\\ne b}}(a+b)\n$$\nhas at least $f(n)$ distinct prime factors. Erdős asked whether\n$$\n\\frac{f(n)}{\\log n}\\to\\infty.\n$$\nThe answer is yes.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
  "solution_events": [
    {
      "id": "vibemathed:erdos-problem-126-prime-divisors-of-pairwise-sums:event",
      "problem_id": "vibemathed:erdos-problem-126-prime-divisors-of-pairwise-sums",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-08-28T00:00:00.000Z",
      "title": "Github",
      "summary": "For\n$$\nf(n)=\\min_{|A|=n}\\left|\\left\\{p\\text{ prime}:p\\mid a+b\\text{ for some distinct }a,b\\in A\\right\\}\\right|,\n$$\nAstra formally proves\n$$\n\\frac{f(n)}{\\log n}\\to\\infty.\n$$\n\nThe repository contains substantially stronger proofs. In particular, one verified alternate resolution establishes\n$$\nn\\le 3r^2,\n$$\nwhere $r$ is the number of supporting primes, yielding\n$$\nf(n)\\gg n^{1/2}.\n$$\nOther independent resolutions give exponents $1/3$, $1/5$, and $1/8$. Thus the formal work goes well beyond the qualitative conjecture, although only the limit statement is the registered benchmark theorem.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:erdos-problem-126-prime-divisors-of-pairwise-sums:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-problem-126-prime-divisors-of-pairwise-sums:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-problem-126-prime-divisors-of-pairwise-sums:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-problem-126-prime-divisors-of-pairwise-sums:attempt",
      "problem_id": "vibemathed:erdos-problem-126-prime-divisors-of-pairwise-sums",
      "model": "GPT-6 Astra (pre-release)",
      "provider": "OpenAI",
      "attempted_at": "2026-08-28T00:00:00.000Z",
      "human_collaborators": [
        "Tom Adamczewski"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "GPT-6 Astra autonomously solved the Formal Conjectures benchmark statement and wrote the Lean proofs, with no human seeing or steering the proof search. Remarkably, four independent successful runs produced polynomial lower bounds by apparently three distinct elementary arguments. The primary proof gives $f(n)\\gg n^{1/8}$; alternate proofs give $f(n)\\gg n^{1/3}$, $f(n)\\gg n^{1/2}$, and $f(n)\\gg n^{1/5}$. Only the weaker limit $f(n)/\\log n\\to\\infty$ is advertised and Comparator-checked as the benchmark theorem.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "erdosproblems.com/126: status and Thomas Bloom's proof exposition",
          "url": "https://www.erdosproblems.com/126",
          "kind": "problem-record",
          "license": null
        },
        {
          "label": "Challenge.lean: the compared statement",
          "url": "https://github.com/tadamcz/erdos126/blob/main/Challenge.lean",
          "kind": "lean-statement",
          "license": null
        },
        {
          "label": "Solution.lean and the Lean development",
          "url": "https://github.com/tadamcz/erdos126/blob/main/Solution.lean",
          "kind": "lean-proof",
          "license": null
        },
        {
          "label": "Epoch AI, Announcing FrontierMath Erdős (1 September 2026)",
          "url": "https://epoch.ai/latest/announcing-frontiermath-erdos",
          "kind": "announcement",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "Github",
          "url": "https://github.com/tadamcz/erdos126",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #126: prime divisors of pairwise sums",
          "url": "https://vibemathed.com/problem/erdos-problem-126-prime-divisors-of-pairwise-sums",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": 247,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-problem-126-prime-divisors-of-pairwise-sums:verification",
      "problem_id": "vibemathed:erdos-problem-126-prime-divisors-of-pairwise-sums",
      "solution_event_id": "vibemathed:erdos-problem-126-prime-divisors-of-pairwise-sums:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Lean-verified. Checked here on 6 September 2026 from a clone of tadamcz/erdos126 at abd4239: 7,866 lines of Lean, zero sorry outside the statement stubs, zero axiom declarations, no native_decide, unsafe or implemented_by; Comparator configuration present and CI runs it with only propext, Quot.sound and Classical.choice. The statement is copied verbatim from Formal Conjectures' ErdosProblems/126.lean at commit 488aade2. Only the qualitative limit f(n)/log n -> infinity is the compared theorem; the polynomial bounds (exponents 1/8, 1/3, 1/2, 1/5 across four runs) are stronger internal results, and erdosproblems.com's page records the n^(1/2) one. erdosproblems.com, the field's own record, marks the problem PROVED (LEAN) with a proof exposition by Thomas Bloom, which is why this is Resolved rather than Candidate: the canonical tracker has accepted it.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #126: prime divisors of pairwise sums",
          "url": "https://vibemathed.com/problem/erdos-problem-126-prime-divisors-of-pairwise-sums",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Github",
          "url": "https://github.com/tadamcz/erdos126",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "Github",
      "url": "https://github.com/tadamcz/erdos126",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #126: prime divisors of pairwise sums",
      "url": "https://vibemathed.com/problem/erdos-problem-126-prime-divisors-of-pairwise-sums",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-problem-126-prime-divisors-of-pairwise-sums:signal:replay_ready",
      "problem_id": "vibemathed:erdos-problem-126-prime-divisors-of-pairwise-sums",
      "kind": "replay_ready",
      "reason": "A public source and a non-unreviewed verification label support replay triage.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}