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PreprintCC BY 4.0arXiv

Carmichael numbers of the form 2n pm+1 with odd exponent m: refinements of a theorem of Florian Luca, a general independence lemma, and a computational study

  • Pagdame Tiebekabe
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Abstract

Florian Luca devoted the last paper of his life to the set 𝒦 of odd positive integers k such that 2n k+1 is a Carmichael number for some positive integer n. He proved that if m≥ 5 is fixed and odd, then there are only finitely many primes p with pm∈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=2n pm+1 with m odd satisfies n<22·107(m+1)2m2(log p)2, and if p>m the much sharper bound n<25m4log p holds; for p>500m5log m we further record the uniform bounds ω(N)≤ 26m7 on the number of prime factors. As an application, the number of such Carmichael numbers up to x is Om(x1/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≥3 is odd and coprime to b, p>m is prime and N=bn pm+1 is Carmichael, then bn pm and ba pβ are multiplicatively independent whenever ba pβ+1 is a prime factor of N. Third, we report an exhaustive computation showing that the only Carmichael number 2n pm+1≤ 1018 with odd 3≤ m≤ 37 is 1729=26· 33+1, that no such number exists with odd 5≤ m≤ 37 (extended to 1021 for 5≤ m≤ 15), and that the only example with even 4≤ m≤ 36 is 46657=26· 36+1=13· 37· 97, where 36=272 is the square of the smallest element of 𝒦.

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Cite

BibTeXRIS
BibTeX
@misc{tiebekabe2026carmichael,
  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}},
  author = {Pagdame Tiebekabe},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25104},
  eprint = {2609.25104},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25104},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - 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
AU  - Pagdame Tiebekabe
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.25104
UR  - https://arxiv.org/abs/2609.25104
AB  - Florian Luca devoted the last paper of his life to the set 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≥ 5 is fixed and odd, then there are only finitely many primes p with p^m∈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<2^2·10^7(m+1)^2m^2(log p)^2, and if p>m the much sharper bound n<25m^4log p holds; for p>500m^5log m we further record the uniform bounds ω(N)≤ 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≥3 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≤ 10^18 with odd 3≤ m≤ 37 is 1729=2^6· 3^3+1, that no such number exists with odd 5≤ m≤ 37 (extended to 10^21 for 5≤ m≤ 15), and that the only example with even 4≤ m≤ 36 is 46657=2^6· 3^6+1=13· 37· 97, where 3^6=27^2 is the square of the smallest element of 𝒦.
ER  -