{
  "abstract": "Let $b, p\\in {\\mathbb Z}$ with $b\\ge 2$ and $p\\ge 3$ a prime. If $b^{p-1}\\equiv 1 \\pmod{p^2}$, then $p$ is called a {\\em generalized Wieferich prime base $b$}, or more succinctly, a {\\em base-$b$ Wieferich prime}. When $b=2$, $p$ is also known simply as a Wieferich prime. We say that a monic polynomial $f(x)\\in {\\mathbb Z}[x]$ is {\\em monogenic} if $f(x)$ is irreducible over ${\\mathbb Q}$ and $\\{1,θ,θ^2,\\ldots,θ^{°(f)-1}\\}$ is a basis for the ring of integers of ${\\mathbb Q}(θ)$, where $f(θ)=0$. Recently, necessary and sufficient conditions for the monogenicity of the trinomials $x^{2n}+bx^n+b$ were given that included base-$b$ Wieferich prime congruences, and also congruences involving a certain Lucas sequence. In this article, we prove a similar result for a different class of trinomials that does not rely on any congruence condition involving a Lucas sequence. Furthermore, we extend this result to a related class of $N$-nomials, for any $N\\ge 4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13904",
  "author_total": 1,
  "authors": [
    {
      "name": "Lenny Jones"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ctsfjej75jhzd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13904",
  "fulltext_url": null,
  "id": "ctsfjej75jhzd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13904",
  "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": "W7213331060",
  "pdf_url": "https://arxiv.org/pdf/2609.13904",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ctsfjej75jhzd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13904",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13904"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Generalized Wieferich primes and monogenic polynomials",
  "title_text": "Generalized Wieferich primes and monogenic polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/ctsfjej75jhzd",
  "venue": "arXiv",
  "versions": []
}