{
  "abstract": "Let $k$ be a positive integer, and consider the sequences of positive rationals with $x_0\\in\\N$ and $x_{n+1}=k/(x_0+x_1+\\dots+x_n)$. Write $x_n=a_n/b_n$ in lowest terms. We show that there is a rational constant $c>0$ such that $c\\,a_n+b_n$ is a perfect power for every such sequence and every $n\\ge2$ if and only if $k=h^2$ and every prime factor of $h$ is congruent to $3$ modulo $4$; in that case $c=2/h$ and the powers are squares. The proof rests on the observation that $t_n=(x_0+\\dots+x_n)/h$ satisfies $t_{n+1}=t_n+1/t_n$, a recursion under which reduced fractions never cancel. This yields an exact formula for $c\\,a_n+b_n$, shows that along any single sequence each prime factor of $h$ spoils at most one term, and leads to two generalizations. For arbitrary $k$ the invariant $b_n^2-\\frac4k a_n^2$ is always a rational square, and it is always an integer square exactly when the primes dividing the square part of $k$ satisfy an inertness condition in $\\Q(\\sqrt{-k})$. For the recursions $x_{n+1}=h^2/(x_0+\\dots+x_n+nμh)$ the role of the Gaussian integers is played by the quadratic fields $\\Q(\\sqrt{μ^2-4})$, which include $\\Q(\\sqrt{-3})$ and every real quadratic field.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00765",
  "author_total": 3,
  "authors": [
    {
      "name": "Mateo Matijasevick"
    },
    {
      "name": "Santiago Rodríguez"
    },
    {
      "name": "Gregorio Salazar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dksfwzc1kwf1q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00765",
  "fulltext_url": null,
  "id": "dksfwzc1kwf1q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00765",
  "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",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7216550722",
  "pdf_url": "https://arxiv.org/pdf/2610.00765",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dksfwzc1kwf1q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00765",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00765"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Perfect squares in reciprocal-sum sequences and primes that are inert in quadratic fields",
  "title_text": "Perfect squares in reciprocal-sum sequences and primes that are inert in quadratic fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/dksfwzc1kwf1q",
  "venue": "arXiv",
  "versions": []
}