{
  "abstract": "For coprime integers $a$ and $b$, it is known that exactly one of the two Diophantine equations $$ ax+by\\ =\\ \\frac{(a-1)(b-1)}{2} \\qquad\\text{and}\\qquad 1+ax+by\\ =\\ \\frac{(a-1)(b-1)}{2} $$ admits a nonnegative integer solution, and that this solution is unique. We first generalize this result by replacing the right-hand side with an arbitrary integer $m$ and its complement $ab-a-b-m$. This framework enables us to study the existence and uniqueness of nonnegative integer solutions to $$ ax+by\\ =\\ \\frac{(a-1)(b-1)}{k} \\qquad\\text{and}\\qquad 1+ax+by\\ =\\ \\frac{(a-1)(b-1)}{k}, $$ where $k$ is a fixed positive integer. We then obtain explicit results when $a$ and $b$ are consecutive Fibonacci numbers. Finally, we examine the original pair of equations in several particular settings, including when $b\\equiv \\pm1\\mod a$, when $b$ is replaced by a higher power, and when the parameters are squared.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08728",
  "author_total": 5,
  "authors": [
    {
      "name": "Hung Viet Chu"
    },
    {
      "name": "Dongho Kim"
    },
    {
      "name": "Theodore Koch"
    },
    {
      "name": "Steven J. Miller"
    },
    {
      "name": "Anh Viet Nguyen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5sz19cvq4ry1r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08728",
  "fulltext_url": null,
  "id": "5sz19cvq4ry1r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08728",
  "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.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7212028413",
  "pdf_url": "https://arxiv.org/pdf/2609.08728",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5sz19cvq4ry1r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08728",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08728"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Generalizing a Pair of Diophantine Equations",
  "title_text": "Generalizing a Pair of Diophantine Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/5sz19cvq4ry1r",
  "venue": "arXiv",
  "versions": []
}