{
  "abstract": "We study three innocent-looking recurrences modulo $3$, each built from squares of two earlier terms. One of them turns out to be secretly $\\textit{linear}$, and we can compute its periods exactly. One is a $\\textit{permutation}$ hiding a small, perfectly orderly piece inside a large, random-looking one. The third looks thoroughly random, but it still has a few provable features. Along the way we meet open problems that need nothing beyond a first course in abstract algebra to state. This paper is a starting point, not the final word in any sense.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34300",
  "author_total": 1,
  "authors": [
    {
      "name": "Steven Finch"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8kesn6fkr17md.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34300",
  "fulltext_url": null,
  "id": "8kesn6fkr17md",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34300",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7223791644",
  "pdf_url": "https://arxiv.org/pdf/2610.34300",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8kesn6fkr17md.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34300",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34300"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Three Quadratic Recurrences Modulo 3: What We Can Prove and What We Can Guess",
  "title_text": "Three Quadratic Recurrences Modulo 3: What We Can Prove and What We Can Guess",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/8kesn6fkr17md",
  "venue": "arXiv",
  "versions": []
}