{
  "abstract": "We present an elementary but powerful method for proving that a two-variable equation has no integer solutions. The method produces quadratic identities which every integer solution must satisfy. Once these identities are established, a standard quadratic reciprocity argument finishes the proof. We illustrate the power of the method by resolving $25$ out of $30$ two-variable equations of size up to $34$ for which the problem of integer solubility remained open. We also resolve the remaining $5$ equations by deeper methods, and further determine the integer solubility of all two-variable equations of size up to $35$ and of length up to $12$, with two exceptions. Before this work, there were $211$ equations in this category that resisted the previously known methods, and we proved that $209$ of them have no integer solutions, with all proofs formalized in Lean. In addition, we show that our method can be used to solve certain two-variable equations whose integer solution set is non-empty.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18916",
  "author_total": 2,
  "authors": [
    {
      "name": "Bogdan Grechuk"
    },
    {
      "name": "Eugene Go"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/69cq8fv7xhnwz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18916",
  "fulltext_url": null,
  "id": "69cq8fv7xhnwz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18916",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7222364261",
  "pdf_url": "https://arxiv.org/pdf/2610.18916",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/69cq8fv7xhnwz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.18916",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18916"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2608|Algebraic Geometry and Number Theory",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Certifying non-existence of integer points on curves",
  "title_text": "Certifying non-existence of integer points on curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/69cq8fv7xhnwz",
  "venue": "arXiv",
  "versions": []
}