{
  "abstract": "For an arbitrary prime $p$ and arbitrary positive integers $r$ and $d_{1}, \\ldots, d_{r}$ with $d_{r} > \\cdots > d_{1}$, we show that $x^{d_{r}} + x^{d_{r-1}} + \\cdots + x^{d_{2}} + x^{d_{1}} + p$ removed of all of its monic irreducible reciprocal factors is irreducible. As a consequence, we establish a number of results related to a conjecture of Charles Nicol that, over the rationals, the sum of two cyclotomic polynomials $Φ_{n}(x) + Φ_{m}(x)$ is a product of cyclotomic polynomials times either $2$ or an irreducible polynomial, where $n$ and $m$ are integers exceeding $1$. We also establish similar results for $Φ_{n}(x)Φ_{m}(x)+1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05568",
  "author_total": 3,
  "authors": [
    {
      "name": "Michael Filaseta"
    },
    {
      "name": "Lilit Martirosyan"
    },
    {
      "name": "London Cameron Swan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dt0pkbybsr4r2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05568",
  "fulltext_url": null,
  "id": "dt0pkbybsr4r2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05568",
  "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": "W7220681658",
  "pdf_url": "https://arxiv.org/pdf/2610.05568",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dt0pkbybsr4r2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05568",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05568"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the factorization of $x^{d_{r}} + x^{d_{r-1}} + \\cdots + x^{d_{2}} + x^{d_{1}} + p$ and a conjecture of C. Nicol",
  "title_text": "On the factorization of x^d_r + x^d_r−1 + ⋯ + x^d_2 + x^d_1 + p and a conjecture of C. Nicol",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/dt0pkbybsr4r2",
  "venue": "arXiv",
  "versions": []
}