{
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  "problem_id": "vibemathed:transcendence-in-the-affine-case-of-erdos-problem-270",
  "title": "Transcendence in the affine case of Erdős Problem 270",
  "canonical_statement": "For integers $a\\geq1$ and $b\\geq1-a$, the series $C_{a,b}=\\sum_{n=1}^{\\infty} n!/((a+1)n+b)!$ is transcendental. Equivalently, the series in Erdős Problem 270 is transcendental whenever $f(n)=an+b$ is a positive integer-valued affine function.",
  "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:transcendence-in-the-affine-case-of-erdos-problem-270:event",
      "problem_id": "vibemathed:transcendence-in-the-affine-case-of-erdos-problem-270",
      "type": "proved",
      "status": "candidate",
      "occurred_at": "2026-08-22T00:00:00.000Z",
      "title": "Transcendence in the Affine Case of Erdős Problem 270",
      "summary": "The manuscript claims $C_{a,b}$ is transcendental for every $a\\ge1$ and $b\\ge1-a$, settling the positive integer-valued affine subclass of Erdős Problem 270. Two pieces of context matter. Problem 270 as Erdős and Graham posed it, for every $f(n)\\to\\infty$, was already answered no by Crmarić and Kovač in 2025: for any $\\alpha>0$ some such $f$ makes the series sum to $\\alpha$. What survives is the non-decreasing case, and the affine family sits inside it. Separately, the checkable parts here were already known - the short irrationality proof for $C_{1,0}$ is Crmarić and Kovač's, posted by Kovač on the Erdős Problems forum in July 2026 and credited in the repository, and base-case transcendence follows a 2023 MathOverflow argument. The new content is the extension to the whole affine family, which is the part with neither formalization nor review.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:transcendence-in-the-affine-case-of-erdos-problem-270:attempt"
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  "attempts": [
    {
      "id": "vibemathed:transcendence-in-the-affine-case-of-erdos-problem-270:attempt",
      "problem_id": "vibemathed:transcendence-in-the-affine-case-of-erdos-problem-270",
      "model": "GPT-5.6 Sol (Codex)",
      "provider": "OpenAI",
      "attempted_at": "2026-08-22T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The author's account, given to this site on submission rather than in the manuscript: OpenAI Codex independently rediscovered the elementary denominator argument, connected Crmarić and Kovač's Gaussian integral representation with a MathOverflow Siegel-Shidlovsky argument to obtain base-case transcendence, and developed the claimed extension to all positive affine cases using hypergeometric E-functions, Euler-operator reductions and a formal-at-infinity resonance argument. It located the relevant results of Salikhov, Salikhov-Viskina and Beukers, drafted the manuscript, and produced most of the Lean formalization; further AI reviews identified gaps and prompted revisions. The manuscript's own disclosure is a single line - \"This manuscript was written and checked using generative AI\" - which names no model and describes only writing and checking. The tier here follows the detailed account because the submitter is the author, but a reader following the source link will not find it there.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Kovač's earlier elementary irrationality proof",
          "url": "https://www.erdosproblems.com/forum/thread/270#post-7469",
          "kind": "independent",
          "license": null
        },
        {
          "label": "Crmarić and Kovač, On the irrationality of certain super-polynomially decaying series - answers Problem 270 as posed",
          "url": "https://arxiv.org/abs/2504.18712",
          "kind": "independent",
          "license": null
        },
        {
          "label": "Erdős Problem #270",
          "url": "https://www.erdosproblems.com/270",
          "kind": "problem-record",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "Transcendence in the Affine Case of Erdős Problem 270",
          "url": "https://github.com/clambro/erdos-270-transcendence",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Transcendence in the affine case of Erdős Problem 270",
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          "kind": "source-assertion",
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      "independence": "unknown"
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  "verifications": [
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      "id": "vibemathed:transcendence-in-the-affine-case-of-erdos-problem-270:verification",
      "problem_id": "vibemathed:transcendence-in-the-affine-case-of-erdos-problem-270",
      "solution_event_id": "vibemathed:transcendence-in-the-affine-case-of-erdos-problem-270:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Read here on 24 August 2026 at github.com/clambro/erdos-270-transcendence. The elementary irrationality theorem for $a\\ge1$, $0\\le b\\le a$ is unconditional, and the five proof modules total 977 lines with no sorry, no admit, no declared axiom and no native_decide on Lean 4.33.1. The general transcendence theorem is not formalized. Salikhov, Salikhov-Viskina, Beukers, the Levelt-Turrittin decomposition and the step from the resonance calculation to functional minimality are passed as explicit Lean hypotheses rather than hidden behind axiom declarations, which is the honest construction; but for the novel range $b>a$ the hypothesis BeyondStripInput.pair is the conclusion itself, so the formalization lends the new claim no independent weight. Lean was not compiled here. The manuscript is unrefereed, self-published in a GitHub repository, and two days old at review.",
      "sources": [
        {
          "label": "VibeMathed: Transcendence in the affine case of Erdős Problem 270",
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          "label": "Transcendence in the Affine Case of Erdős Problem 270",
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      "label": "Transcendence in the Affine Case of Erdős Problem 270",
      "url": "https://github.com/clambro/erdos-270-transcendence",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Transcendence in the affine case of Erdős Problem 270",
      "url": "https://vibemathed.com/problem/transcendence-in-the-affine-case-of-erdos-problem-270",
      "kind": "source-assertion",
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    {
      "id": "vibemathed:transcendence-in-the-affine-case-of-erdos-problem-270:signal:verify_now",
      "problem_id": "vibemathed:transcendence-in-the-affine-case-of-erdos-problem-270",
      "kind": "verify_now",
      "reason": "The source labels this result as a candidate; verification should precede reuse.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
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  "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"
}