Carmichael numbers and least common multiples of p-1

2024/09/24 by Thomas Wright, Wright, Thomas
Mathematics · #11A51 #Advanced Mathematical Identities #Advanced Mathematical Theories #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2409.16397

openalex publication_date 2024/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a Carmichael number n with prime factors p₁,⋯,p_m, define K=GCD[p₁-1,⋯,p_m-1], and let C_ν(X) denote the number of 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 even natural number ν, C_ν(X)≥ X^{1-(2+o(1))(logloglog log X)/(logloglog X)}. This is a departure from standard constructions of Carmichael numbers, which generally require K to grow along with n.

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