{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-707-sidon-perfect-difference",
  "title": "Erdős Problem #707: Sidon Sets and Perfect Difference Sets",
  "canonical_statement": "Erdős conjectured, in over a dozen papers spanning 1976 to 1997 and with a 1000 dollars prize attached, that every finite Sidon set extends to a perfect difference set modulo $p^2+p+1$ for some prime $p$. Alexeev and Mixon establish that $\\{1,2,4,8\\}$ is a counterexample - and discovered along the way that Marshall Hall, Jr. had published a different counterexample three decades before Erdős first posed the problem, unnoticed by the community for half a century.",
  "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-707-sidon-perfect-difference:event",
      "problem_id": "vibemathed:erdos-707-sidon-perfect-difference",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2025-10-22T00:00:00.000Z",
      "title": "arXiv",
      "summary": "Hall's 1947 counterexample predates the problem itself; this paper's counterexample is independent, smaller, and Lean-certified.",
      "ai_contribution": "ai_assisted",
      "attempt_ids": [
        "vibemathed:erdos-707-sidon-perfect-difference:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-707-sidon-perfect-difference:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-707-sidon-perfect-difference:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-707-sidon-perfect-difference:attempt",
      "problem_id": "vibemathed:erdos-707-sidon-perfect-difference",
      "model": "ChatGPT (GPT-5)",
      "provider": "OpenAI",
      "attempted_at": "2025-10-22T00:00:00.000Z",
      "human_collaborators": [
        "Boris Alexeev",
        "Dustin G. Mixon"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The mathematics is the humans'; the paper is candid that LLMs failed at the two things they are usually praised for here - they never located Hall's paywalled prior solution, and \"even after we knew what exactly to prove, it couldn't help us close the gap.\" What ChatGPT did do: write the complete Lean formalization of both counterexamples (\"we decided to vibe code the whole proof... about a week... somehow it succeeded\"). Earlier versions of the paper listed ChatGPT and Lean as authors until arXiv policy required their removal.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2510.19804",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #707: Sidon Sets and Perfect Difference Sets",
          "url": "https://vibemathed.com/problem/erdos-707-sidon-perfect-difference",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-707-sidon-perfect-difference:verification",
      "problem_id": "vibemathed:erdos-707-sidon-perfect-difference",
      "solution_event_id": "vibemathed:erdos-707-sidon-perfect-difference:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Both Hall's and the new counterexample are formalized and kernel-checked in Lean, with the formalization written by ChatGPT and audited by the authors; erdosproblems.com marks the problem disproved with the proof verified in Lean.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #707: Sidon Sets and Perfect Difference Sets",
          "url": "https://vibemathed.com/problem/erdos-707-sidon-perfect-difference",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2510.19804",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2510.19804",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #707: Sidon Sets and Perfect Difference Sets",
      "url": "https://vibemathed.com/problem/erdos-707-sidon-perfect-difference",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-707-sidon-perfect-difference:signal:replay_ready",
      "problem_id": "vibemathed:erdos-707-sidon-perfect-difference",
      "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"
}