{
  "abstract": "A cyclic number is a positive integer $n$ such that $\\gcd(n, φ(n)) = 1$. Erdős showed that the number of cyclic numbers up to $x$, $C(x)$, has $C(x) \\sim e^{-γ}x/\\log_3(x)$ asymptotically as $x$ tends to infinity. We define Wieferich numbers to be odd, positive integers $n$ such that $(n, (2^{\\ell(n)}-1)/n) > 1$, where $\\ell(n)$ is the smallest positive integer with $2^{\\ell(n)} \\equiv 1 \\mod n$. Franco and Pomerance showed that non-Wieferich numbers have density zero among the odd integers. We show that the counting function of $n \\leq x$ in the symmetric difference of non-Wieferich numbers and cyclic numbers is $O(x/(\\log_3(x))^2)$, under a mild assumption on the counting function of Wieferich primes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28685",
  "author_total": 1,
  "authors": [
    {
      "name": "Casia Siegel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2y9bphnfz0stf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28685",
  "fulltext_url": null,
  "id": "2y9bphnfz0stf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28685",
  "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",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7223249973",
  "pdf_url": "https://arxiv.org/pdf/2610.28685",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2y9bphnfz0stf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.28685",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28685"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "An Asymptotic for Non-Wieferich Numbers",
  "title_text": "An Asymptotic for Non-Wieferich Numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/2y9bphnfz0stf",
  "venue": "arXiv",
  "versions": []
}