{
  "abstract": "Define an $(a,a)$-Carmichael number to be a squarefree natural number $n$ such that $p\\mid n$ implies $p-a\\mid n-a$. For such a number $n$ with prime factors $p_1,\\cdots,p_m$, define $$K=GCD[p_1-a,\\cdots,p_m-a],$$ and let $C_ν(X,a)$ denote the number of $(a,a)$-Carmichael numbers up to $X$ such that $K=ν$. Assuming a strong conjecture on the first prime in an arithmetic progression, we prove that for any integer $a$ and for any natural number $ν$ with $(ν,a)=1$ and $a$ and $ν$ having opposite parity, $$C_ν(X,a)\\geq X^{1-(2+o(1))\\frac{\\log\\log\\log \\log X}{\\log\\log\\log X}}.$$ This is a departure from many traditional constructions of Carmichael numbers, which generally require $K$ to grow along with $n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.02738",
  "author_total": 1,
  "authors": [
    {
      "name": "Thomas Wright"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/96a465zv87ff8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.02738",
  "fulltext_url": null,
  "id": "96a465zv87ff8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.02738",
  "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": "W7167601563",
  "pdf_url": "https://arxiv.org/pdf/2607.02738",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/96a465zv87ff8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2607.02738",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.02738"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "$(a,a)$-Carmichael numbers and greatest common divisors of $p-a$",
  "title_text": "(a,a)-Carmichael numbers and greatest common divisors of p−a",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/96a465zv87ff8",
  "venue": "arXiv",
  "versions": []
}