{
  "abstract": "Let $Φ_r(a,b)$ denote the homogeneous cyclotomic value. For sufficiently large $A$, uniformly for $1/2\\leq B\\leq A$, $r\\geq3$ and $Z\\geq2$, the number of coprime pairs in $(A,2A]\\times(B,2B]$ whose greatest prime factor $P(Φ_r(a,b))$ is at most $Z$ is $O(AB/φ(r)+\\sqrt{r/φ(r)}\\,π(Z;r,1))$, where $π(Z;r,1)$ counts primes at most $Z$ congruent to $1$ modulo $r$. If $n=n(A)\\to\\infty$ through positive integers and $1\\leq H(A)=o(\\log A)$, then $P(a^{2^n}+1)>2^nAH(A)$ for almost all $a\\in(A,2A]$, and $P(a^{2^n}+b^{2^n})>2^nA^2H(A)$ for almost all coprime pairs in $(A,2A]^2$, with no further restriction on $n$. For every fixed $α>0$ and every $r\\geq A^α$, pairwise distinct primitive prime divisors of $Φ_r(a)$ can be selected for all but $O_α(A^{1-c_α})$ bases in $(A,2A]$, where $c_α>0$. For coprime positive integers $a,b$ with $\\max(a,b)\\geq2$ and $r\\geq3$, put $C=Φ_r(a,b)$ and $S=φ(r)^2\\log\\max(a,b)$. Uniformly as $S\\to\\infty$, $\\max_{p\\mid C}p^{v_p(C)}\\geq(5/16-o(1))S$. For every positive function $h(S)\\to\\infty$, $\\sum_{p\\mid C,\\;p^{v_p(C)}>S/h(S)}v_p(C)\\log p =(1-o(1))\\log C$ uniformly in the same limit.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00075",
  "author_total": 1,
  "authors": [
    {
      "name": "Wei Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fyyf81jvsy9w3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00075",
  "fulltext_url": null,
  "id": "fyyf81jvsy9w3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00075",
  "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.RA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00075",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fyyf81jvsy9w3.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00075",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00075"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Primary components and large prime factors of cyclotomic values",
  "title_text": "Primary components and large prime factors of cyclotomic values",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/fyyf81jvsy9w3",
  "venue": "arXiv",
  "versions": []
}