{
  "abstract": "Consider the polynomial $f=x^N+y^N-z^N$ where $x,\\,y$ and $z$ are positive integers and $N \\ge 3$ is an integer. By Fermat's Last Theorem, $f$ is never zero so that its reciprocal, $1/f$, has no singularities. We therefore study the finite sum of the reciprocal: $S(m,N)=\\sum_{x=1}^m\\sum_{y=1}^m\\sum_{z=1}^{m}\\frac{1}{f}$. The terms $1/f$ can be positive, negative and their magnitude is less than unity. A key observation is that $S(m,N)$ can be split into two convenient parts: a dominant contribution $D(m,N)$ that has a simple analytical expression and a remainder $R(m,N)$ which is more complicated but negligible compared to $D(m,N)$. Therefore, $S(m,N)$ is almost identical to $D(m,N)$. The analytical expression for $D(m,N)$ is $(2\\,m-1)\\,H_m^{(N)}$ where $H_m^{(N)}=\\sum_{x=1}^m\\frac{1}{x^N}$ approaches quickly the Riemann zeta function $ζ(N)$ as $m$ increases. Therefore, the original sum $S(m,N)$ has a simple expression: it is basically linear in $m$ with slope equal to $2\\,ζ(N)$. Its linear behavior is not an asymptotic result; plots of $S(m,N)$ vs. $m$ for different $N$ show a straight line starting at $m=1$. $S(m,N)$ deviates slightly from a straight line over a small interval $8\\le m\\le 12$ for the case $N=3$. This slight deviation is due to Fermat near misses where $x^3+y^3-z^3=\\pm 1$ (for $z\\ne x$ and $z\\ne y$); these create a jump in the remainder $R(m,3)$ at $m=9$. We make a numerical and analytical study of the remainder $R(m,N)$. From the numerical analysis, $R(m,N)$ converges for $N\\ge 4$ but it was harder to tell whether $N=3$ converged. An analytical study based on a comparison of $R(m,N)$ to its Cauchy principal value integral, shows that $R(m,3)$ likely diverges logarithmically. It also shows that $R(m,N)$ converges for $N\\ge 4$ in agreement with the numerical analysis. We discuss in the conclusion some interesting questions for future investigation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02112",
  "author_total": 1,
  "authors": [
    {
      "name": "Ariel Edery"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/219tj1bn8p3w5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02112",
  "fulltext_url": null,
  "id": "219tj1bn8p3w5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02112",
  "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",
      "hep-th"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7207840822",
  "pdf_url": "https://arxiv.org/pdf/2609.02112",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/219tj1bn8p3w5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:12Z",
      "native_id": "2609.02112",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02112"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "Summing the reciprocal of the polynomial appearing in Fermat's Last Theorem",
  "title_text": "Summing the reciprocal of the polynomial appearing in Fermat's Last Theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/219tj1bn8p3w5",
  "venue": "arXiv",
  "versions": []
}