On the convergence of series of reciprocals of primes related to the Fermat numbers
type: Item
instance of: scholarly article
title: On the convergence of series of reciprocals of primes related to the Fermat numbers
zbMATH Open document ID: 1026.11011
author: Michal Křížek
author: Florian Luca
author: Lawrence Somer
published in: Journal of Number Theory
publication date: 2003-04-02T00:00:00Z
review text: For any integer \(m\geq 0\) let \(F_m\) be the \(m\)th Fermat number. Let us denote by \(P\) the set of all primes \(p\) for which there exists \(m\) such that \(p\) divides \(F_m\). In 1955, \textit{S. W. Golomb} [Math. Scand. 3, 264-274 (1955; Zbl 0067.27503)] asked if the series whose general term is \(1/p\) \((p\in P)\) is convergent. A positive answer is given in this paper. \textit{A. Aigner} [Monatsh. Math. 101, 85-93 (1986; Zbl 0584.10003)] introduced the notion of an elite prime as being a prime number \(p\) such that \(p\) is a quadratic nonresidue modulo \(F_m\) for all but finitely many values of \(m\). Denote by \(E\) the set of all elite primes. It is shown that the series whose general term is \(1/p\) with \(p\in E\) is convergent.
reviewed by: Solomon Marcus
Mathematics Subject Classification ID: 11A51
Mathematics Subject Classification ID: 11N13
zbMATH DE Number: 1891430
zbMATH Keywords: Fermat numbers
zbMATH Keywords: elite primes
zbMATH Keywords: sums of reciprocals
zbMATH Keywords: density
MaRDI profile type: Publication
full work available at URL: jnth.2002.2782
OpenAlex ID: W2006366105
Wikidata QID: Q56114458
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DOI: 10.1006/JNTH.2002.2782
description: scientific article; zbMATH DE number 1891430