{
  "abstract": "For an integer solution of $y^2=x^3+k$ with $x,y\\ge 1$ and $k\\ne 0$, the integers $x^3$, $|k|$ and $y^2$ form a triple in which one term is the sum of the other two. When $\\gcd(x,y)=1$ this triple is coprime, and the $abc$ conjecture predicts that its quality is asymptotically at most $1$. Following Müller, Taktikos and de Weger, we factor this quality through the product $P=xy|k|$ into the approximation gain $G_a=\\log\\max(x^3,y^2)/\\log P$ and the power gain $G_p=\\log P/\\log\\rad(xy|k|)$, which is at least $1$. We show that the weak form of Hall's conjecture, which asks that $|k|>x^{1/2-δ}$ for every $δ>0$ outside finitely many solutions, is equivalent to the asymptotic bound $G_a\\le 1$. More generally, for $6/11\\leκ<6/5$, an asymptotic bound $G_a\\leκ$ gives $|k|>x^{3/κ-5/2-δ}$ for every $δ>0$ outside finitely many solutions, and for $3/4\\leκ<6/5$ the two statements are equivalent. Since the approximation gain never exceeds the quality, an $abc$ inequality with exponent $1\\leκ<6/5$ gives the same bound for primitive solutions. The case $κ=1$ extends to all solutions and recovers the classical implication from the $abc$ conjecture to weak Hall. We also relate small values of $|k|$ to large partial quotients of $\\sqrt x$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35252",
  "author_total": 2,
  "authors": [
    {
      "name": "R. Laniewski"
    },
    {
      "name": "K. Müller"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ecqfr9ast2055.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35252",
  "fulltext_url": null,
  "id": "ecqfr9ast2055",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35252",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214715679",
  "pdf_url": "https://arxiv.org/pdf/2609.35252",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ecqfr9ast2055.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.35252",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35252"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Weak Hall's conjecture, approximation gains, and continued fractions",
  "title_text": "Weak Hall's conjecture, approximation gains, and continued fractions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/ecqfr9ast2055",
  "venue": "arXiv",
  "versions": []
}