There are no Carmichael numbers of the form \(2^np+1\) with \(p\) prime
type: Item
instance of: scholarly article
title: There are no Carmichael numbers of the form \(2^np+1\) with \(p\) prime
zbMATH Open document ID: 1504.11013
author: Florian Luca
author: Adel Alahmadi
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2022-10-12T00:00:00Z
review text: A Sierpinski number is an odd positive integer \(k\) such that \(2^nk+1\) is composite for every positive integer \(n\). A Carmichael number is an odd integer \(N\) such that it is composite but \(a^N\equiv a\pmod{N}\) for every \(a\). This paper relates both concepts by considering the set \[ \mathcal{K}=\{k \text{ odd}: \{2^nk+1\}_{n\geq 0} \text{ contains some Carmichael number}\} \] In particular it is proved that all members of \(\mathcal{K}\) are in fact composite. Note that this is equivalent to saying that there is no Carmichael number of the form \(2^np+1\) with \(p\) an odd prime.
Mathematics Subject Classification ID: 11A51
zbMATH DE Number: 7600088
zbMATH Keywords: Carmichael numbers
zbMATH Keywords: Sierpinski numbers
reviewed by: Antonio M. Oller Marcén
MaRDI profile type: Publication
full work available at URL: crmath.393
OpenAlex ID: W4300978941
cites work: There are infinitely many Carmichael numbers
cites work: Sierpiński and Carmichael numbers
cites work: Carmichael numbers in the sequence $(2^n k+1)_{n\geq 1}$
cites work: On the density of odd integers of the form \((p-1)2^{-n}\) and related questions
cites work: Primes in intervals of bounded length
cites work: The Impossibility of Certain Types of Carmichael Numbers
DOI: 10.5802/CRMATH.393
description: scientific article; zbMATH DE number 7600088