{
  "abstract": "Florian Luca devoted the last paper of his life to the set $\\mathcal K$ of odd positive integers $k$ such that $2^n k+1$ is a Carmichael number for some positive integer $n$. He proved that if $m\\ge 5$ is fixed and odd, then there are only finitely many primes $p$ with $p^m\\in\\mathcal K$. In this note, written as a small tribute to his memory, we extend his result in three directions, using only his published work as our reference base. First, we extract from his arguments explicit quantitative bounds: every Carmichael number $N=2^n p^m+1$ with $m$ odd satisfies $n<2^{2\\cdot10^7(m+1)^2m^2(\\log p)^2}$, and if $p>m$ the much sharper bound $n<25m^4\\log p$ holds; for $p>500m^5\\log m$ we further record the uniform bounds $ω(N)\\le 26m^7$ on the number of prime factors. As an application, the number of such Carmichael numbers up to $x$ is $O_m(x^{1/m})$. Second, we generalize the multiplicative independence step of Luca's proof from base $2$ to an arbitrary even base $b$: if $b$ is even, $m\\ge3$ is odd and coprime to $b$, $p>m$ is prime and $N=b^n p^m+1$ is Carmichael, then $b^n p^m$ and $b^a p^β$ are multiplicatively independent whenever $b^a p^β+1$ is a prime factor of $N$. Third, we report an exhaustive computation showing that the only Carmichael number $2^n p^m+1\\le 10^{18}$ with odd $3\\le m\\le 37$ is $1729=2^6\\cdot 3^3+1$, that no such number exists with odd $5\\le m\\le 37$ (extended to $10^{21}$ for $5\\le m\\le 15$), and that the only example with even $4\\le m\\le 36$ is $46657=2^6\\cdot 3^6+1=13\\cdot 37\\cdot 97$, where $3^6=27^2$ is the square of the smallest element of $\\mathcal K$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25104",
  "author_total": 1,
  "authors": [
    {
      "name": "Pagdame Tiebekabe"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5e43hhvt4cbdk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25104",
  "fulltext_url": null,
  "id": "5e43hhvt4cbdk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25104",
  "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": "W7214082887",
  "pdf_url": "https://arxiv.org/pdf/2609.25104",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5e43hhvt4cbdk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25104",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25104"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Carmichael numbers of the form $2^n p^m+1$ with odd exponent $m$: refinements of a theorem of Florian Luca, a general independence lemma, and a computational study",
  "title_text": "Carmichael numbers of the form 2^n p^m+1 with odd exponent m: refinements of a theorem of Florian Luca, a general independence lemma, and a computational study",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/5e43hhvt4cbdk",
  "venue": "arXiv",
  "versions": []
}