{
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  "ref": "R1372",
  "content_hash": null,
  "slug": "unbounded-integral-points-minimal-elliptic-curves-status-20260801",
  "type": "claim",
  "title": "Current checked status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-27. Large explicit examples and average bounds are known, but the checked sources do not prove unboundedness across global minimal models or an absolute uniform bound.",
  "relevance": "Records the strongest checked neighboring results and the exact remainder future work must settle.",
  "relevance_source": "recorded",
  "body": "A dated independent review on 2026-08-01 checked the structured sources below, the complete visible source discussion, exact-title and equivalent-formulation searches, and the current TheoremDB corpus. On 2026-07-27 all MathOverflow answers and comments were checked. They distinguish large examples on minimal models from changes of variables that manufacture integral points on nonminimal equations. Elkies, arXiv:0709.2908, records elliptic curves with thousands of integral-point pairs and discusses the minimal-model issue. Any new record is finite evidence rather than a proof of unboundedness. Alpöge and Ho, arXiv:1807.03761, prove average moment bounds in families. Average control permits exceptional curves and therefore does not supply a uniform absolute bound. A search also checked recent work on integral points and binary cubic forms, including arXiv:2407.09558. The audit did not verify that its updated version resolves the minimal-model dichotomy, so specialist reconciliation is required. Trap: scaling an equation can turn rational points into integral coordinate pairs while destroying global minimality. Every lower-bound family must certify the model used to define \\(N(E)\\).\n\nA complete resolution must satisfy: For unboundedness, construct or prove the existence of elliptic curves \\(E_r/\\mathbb Q\\) with certified global minimal equations and \\(N(E_r)\\to\\infty\\). For boundedness, give an absolute constant \\(C\\) and prove \\(N(E)\\le C\\) for every elliptic curve over \\(\\mathbb Q\\) on its global minimal integral model.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
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    "kind": "claim",
    "citation": {
      "url": "https://mathoverflow.net/questions/50661/unboundedness-of-number-of-integral-points-on-elliptic-curves",
      "locator": "Dataset references and independent 2026-08-01 status search."
    },
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  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/50661/unboundedness-of-number-of-integral-points-on-elliptic-curves",
    "locator": "Dataset references and independent 2026-08-01 status search."
  },
  "relations": [
    {
      "slug": "R1371",
      "title": "Complete the stated acceptance conditions",
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    {
      "slug": "R1796",
      "title": "Dated status and exact unresolved remainder",
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    {
      "slug": "unbounded-integral-points-minimal-elliptic-curves",
      "title": "unbounded integral points minimal elliptic curves",
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