{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-906",
  "title": "Erdős Problem #906",
  "canonical_statement": "Is there an entire non-zero function $f:\\mathbb{C}\\to \\mathbb{C}$ such that, for any infinite sequence $n_1<n_2<\\cdots$, the set $\\{ z: f^{(n_k)}(z)=0 \\textrm{ for some }k\\geq 1\\}$ is everywhere dense? The literal question is trivial for polynomials, so the claims address the transcendental entire case, in the affirmative.",
  "plain_summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "candidate",
  "solution_events": [
    {
      "id": "vibemathed:erdos-906:event",
      "problem_id": "vibemathed:erdos-906",
      "type": "proved",
      "status": "candidate",
      "occurred_at": "2026-04-25T00:00:00.000Z",
      "title": "erdosproblems.com/906",
      "summary": "Two independent affirmative claims (Adriano's, posted first, and a GPT-5.5 Pro note); Erdős himself wrote in 1982 that the problem had been solved affirmatively long before, without a locatable reference",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:erdos-906:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-906:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-906:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-906:attempt",
      "problem_id": "vibemathed:erdos-906",
      "model": "GPT-5.5 Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-04-25T00:00:00.000Z",
      "human_collaborators": [
        "Przemysław Chojecki"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The probabilistic argument via the cofinite reformulation, using Sodin's Edelman-Kostlan and Offord-type estimates for Gaussian analytic functions, was developed with GPT-5.5 Pro; an independent solution by another contributor was posted first the same day.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "GPT-5.5 Pro note",
          "url": "https://www.ulam.ai/research/erdos906.pdf",
          "kind": "paper",
          "license": null
        },
        {
          "label": "Independent proposed solution",
          "url": "https://github.com/Drill23/erdos-problem-906-proposed-solution",
          "kind": "independent",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "erdosproblems.com/906",
          "url": "https://www.erdosproblems.com/906",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #906",
          "url": "https://vibemathed.com/problem/erdos-906",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-906:verification",
      "problem_id": "vibemathed:erdos-906",
      "solution_event_id": "vibemathed:erdos-906:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "AI screenings reported one minor issue on each claim; no formalization (the required tools are not in mathlib) and no independent expert review; erdosproblems.com still lists the problem open.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #906",
          "url": "https://vibemathed.com/problem/erdos-906",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "erdosproblems.com/906",
          "url": "https://www.erdosproblems.com/906",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "erdosproblems.com/906",
      "url": "https://www.erdosproblems.com/906",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #906",
      "url": "https://vibemathed.com/problem/erdos-906",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-906:signal:verify_now",
      "problem_id": "vibemathed:erdos-906",
      "kind": "verify_now",
      "reason": "The source labels this result as a candidate; verification should precede reuse.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "The source status is candidate and must not be represented as solved.",
    "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"
}