{
  "abstract": "Let $F_n$ and $L_n$ denote the Fibonacci and Lucas numbers, respectively, and consider \\[ E_k:\\qquad y^2=(F_{2k}x+1)(F_{2k+2}x+1)(F_{2k+4}x+1). \\] These elliptic curves arise naturally from the regular Diophantine triples \\[ \\{F_{2k},F_{2k+2},F_{2k+4}\\}. \\] For odd $k$, we exhibit the rational point \\[ Q_k=\\left( -\\frac{F_{k-1}}{L_kF_{k+1}F_{k+2}}, \\frac{F_{2k+1}}{L_kF_{k+1}F_{k+2}} \\right). \\] For every odd $k\\geq 3$, this point is independent of the standard point $P_k=(0,1)$; in particular, $\\operatorname{rank}E_k(\\mathbb{Q})\\geq 2$. Moreover, if $k\\geq 3$ is odd and $\\operatorname{rank}E_k(\\mathbb{Q})=2$, then all integer points on $E_k$ are exactly the points arising from the two known solutions of the Hoggatt-Bergum extension problem. By parametrizing the two conics $L^2-5F^2=\\pm4$ and applying an injective specialization criterion, we also show that the corresponding one-parameter elliptic families have generic ranks $2$ in the odd case and $1$ in the even case. Finally, we discuss computational data and propose the heuristic rank distribution $1/4,1/2,1/4$ for ranks $1,2,3$, respectively, with density zero for rank at least $4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01789",
  "author_total": 1,
  "authors": [
    {
      "name": "Andrej Dujella"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1rs3c5mtycqaw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01789",
  "fulltext_url": null,
  "id": "1rs3c5mtycqaw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01789",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.01789",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1rs3c5mtycqaw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.01789",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01789"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Ranks and integer points on elliptic curves induced by Fibonacci triples",
  "title_text": "Ranks and integer points on elliptic curves induced by Fibonacci triples",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/1rs3c5mtycqaw",
  "venue": "arXiv",
  "versions": []
}