{
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  "source": "vibemath",
  "problem_id": "vibemathed:banks-martin-primitive-sets",
  "title": "The Banks-Martin Conjecture on Primitive Sets",
  "canonical_statement": "Banks and Martin conjectured in 2013 that for a primitive set $A$ and any set $Q$ of primes, the Erdos sum of the members of $A$ composed only of primes in $Q$ is at most the corresponding sum over $Q$ itself. The unrestricted form turned out to be false once $Q$ is allowed to contain $2$; Lichtman proposed a revised form restricted to odd primes. That revised conjecture, long viewed as a unifying master theorem for the area, is proved here.",
  "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:banks-martin-primitive-sets:event",
      "problem_id": "vibemathed:banks-martin-primitive-sets",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-05-01T00:00:00.000Z",
      "title": "arXiv:2605.00301 - Primitive sets and von Mangoldt chains: Erdos Problem #1196 and beyond",
      "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.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:banks-martin-primitive-sets:attempt"
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  "attempts": [
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      "id": "vibemathed:banks-martin-primitive-sets:attempt",
      "problem_id": "vibemathed:banks-martin-primitive-sets",
      "model": "GPT-5.5 Pro (early version)",
      "provider": "OpenAI",
      "attempted_at": "2026-05-01T00:00:00.000Z",
      "human_collaborators": [
        "Boris Alexeev",
        "Kevin Barreto",
        "Yanyang Li",
        "Jared Duker Lichtman",
        "Liam Price",
        "Jibran Iqbal Shah",
        "Quanyu Tang",
        "Terence Tao"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "This paper discloses per theorem rather than in a blanket statement, and this theorem is one of the more modest entries: an early version of GPT-5.5 Pro was used to assist with the initial proof. Elsewhere in the same paper the model's role is larger, with the proof of the Erdos #1196 theorem generated by an autonomous GPT-5.4 Pro run whose transcript is public. Across all of it the authors state that the final proofs were generated and reviewed by them, using the AI-generated proofs as starting points where appropriate. The whole method, Markov chains with von Mangoldt weights, was itself suggested by model output.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
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          "label": "arXiv:2605.00301 - Primitive sets and von Mangoldt chains: Erdos Problem #1196 and beyond",
          "url": "https://arxiv.org/abs/2605.00301",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Banks-Martin Conjecture on Primitive Sets",
          "url": "https://vibemathed.com/problem/banks-martin-primitive-sets",
          "kind": "source-assertion",
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      "solution_event_id": "vibemathed:banks-martin-primitive-sets:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "arXiv preprint, not peer-reviewed. Other results in the same paper were formalized in Lean using Codex and Gauss, but this theorem was not among them.",
      "sources": [
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          "label": "VibeMathed: The Banks-Martin Conjecture on Primitive Sets",
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          "label": "arXiv:2605.00301 - Primitive sets and von Mangoldt chains: Erdos Problem #1196 and beyond",
          "url": "https://arxiv.org/abs/2605.00301",
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      "label": "arXiv:2605.00301 - Primitive sets and von Mangoldt chains: Erdos Problem #1196 and beyond",
      "url": "https://arxiv.org/abs/2605.00301",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Banks-Martin Conjecture on Primitive Sets",
      "url": "https://vibemathed.com/problem/banks-martin-primitive-sets",
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      "generated_at": "2026-09-13T16:28:40.178Z"
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  "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"
}