PreprintCC BY 4.0arXiv
(a,a)-Carmichael numbers and greatest common divisors of p−a
Abstract
Define an (a,a)-Carmichael number to be a squarefree natural number n such that p| n implies p−a| n−a. For such a number n with prime factors p1,⋯,pm, define K=GCD[p1−a,⋯,pm−a], and let Cν(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, Cν(X,a)≥ X1−(2+o(1))(logloglog log X)/(logloglog X). This is a departure from many traditional constructions of Carmichael numbers, which generally require K to grow along with n.
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BibTeX
@misc{wright2026aacarmichael,
title = {{$(a,a)$-Carmichael numbers and greatest common divisors of $p-a$}},
author = {Thomas Wright},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.02738},
eprint = {2607.02738},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.02738},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - (a,a)-Carmichael numbers and greatest common divisors of p−a
AU - Thomas Wright
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2607.02738
UR - https://arxiv.org/abs/2607.02738
AB - Define an (a,a)-Carmichael number to be a squarefree natural number n such that p| n implies p−a| n−a. For such a number n with prime factors p_1,⋯,p_m, define K=GCD[p_1−a,⋯,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)≥ X^1−(2+o(1))(logloglog log X)/(logloglog X). This is a departure from many traditional constructions of Carmichael numbers, which generally require K to grow along with n.
ER -