{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-26",
  "title": "Erdős Problem #26",
  "canonical_statement": "Let $A\\subset\\mathbb{N}$ be infinite. Must there exist some $k\\geq 1$ such that almost all integers have a divisor of the form $a+k$ for some $a\\in A$? The question as posed follows negatively from Davenport–Erdős (1951). The AI result settles Tenenbaum's harder variant, also negatively: there is an infinite $A$ such that for every $k\\geq 1$ the set of multiples of $A+k$ has upper density below $0.34$.",
  "plain_summary": "VibeMathed reports this item as open. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "open",
  "solution_events": [
    {
      "id": "vibemathed:erdos-26:event",
      "problem_id": "vibemathed:erdos-26",
      "type": "disproved",
      "status": "open",
      "occurred_at": "2026-04-06T00:00:00.000Z",
      "title": "erdosproblems.com/26",
      "summary": "The question as posed was implicit in Davenport–Erdős (1951); the AI result settles Tenenbaum's open variant negatively",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:erdos-26:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-26:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-26:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-26:attempt",
      "problem_id": "vibemathed:erdos-26",
      "model": "DeepMind prover agent",
      "provider": "Google DeepMind",
      "attempted_at": "2026-04-06T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "A DeepMind prover agent constructed an infinite set $A$ such that for every $k\\geq 1$ the set of multiples of $A+k$ has upper density less than $0.34$, resolving Tenenbaum's variant of the problem in the negative.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Formalised statement (formal-conjectures)",
          "url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/26.lean",
          "kind": "lean-statement",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "erdosproblems.com/26",
          "url": "https://www.erdosproblems.com/26",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #26",
          "url": "https://vibemathed.com/problem/erdos-26",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-26:verification",
      "problem_id": "vibemathed:erdos-26",
      "solution_event_id": "vibemathed:erdos-26:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "erdosproblems.com marks the problem DISPROVED and documents the DeepMind construction in the page remarks; the variant result is recorded there without a separate formal artifact.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #26",
          "url": "https://vibemathed.com/problem/erdos-26",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "erdosproblems.com/26",
          "url": "https://www.erdosproblems.com/26",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "erdosproblems.com/26",
      "url": "https://www.erdosproblems.com/26",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #26",
      "url": "https://vibemathed.com/problem/erdos-26",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-26:signal:watch_only",
      "problem_id": "vibemathed:erdos-26",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "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"
}