Year: 2026
Type: preprint
Abstract: We prove that no Carmichael number has the form \[ N=(2^mJ_n+1)(2^mJ_{n+1}+1)(2^mJ_{n+2}+1), \] where \[ J_n=\frac{2^n-(-1)^n}{3} \] is the Jacobsthal sequence, \(m,n\ge1\), and all three factors are prime. Assuming that such an \(N\) is Carmichael, Korselt's criterion yields divisibility conditions relating consecutive Jacobsthal numbers to powers of two. Combining these conditions with the Binet formula and a finite case analysis for \(n\le4\), we obtain a contradiction. Consequently, this family contains no Carmichael numbers.
Source: arXiv (Cornell University)
Author: Bensella, Hayat
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222993030