Analogues to Fermat primes related to Pell's equation
type: Item
instance of: scholarly article
title: Analogues to Fermat primes related to Pell's equation
zbMATH Open document ID: 1263.11006
published in: Archiv der Mathematik
publication date: 2010-02-25T00:00:00Z
review text: In this work the author studies the Fermat primes related to Pell equations. He considers the primes of the form \[ p=2x^{2}-1=2dy^{2}+1. \] So it is known that if \((x_{1},y_{1})\) is the positive solution of \(x^{2}-dy^{2}=1\), then the other solutions are \((x_{n},y_{n})\), where \[ x_{n}+y_{n}\sqrt{d}=(x_{1}+y_{1}\sqrt{d})^{n} \] for \(n\geq 1\). Therefore, if \(p_{n}=2x_{n}^{2}-1\) is prime, then \(n=2^{m}\) is a power of \(2\). So there are analogues to the Fermat numbers of the form \(2^{n}+1\).
reviewed by: Ahmet Tekcan
Mathematics Subject Classification ID: 11A41
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 5675007
zbMATH Keywords: Fermat primes
zbMATH Keywords: Pell equation
zbMATH Keywords: modular forms
author: Günter Köhler
MaRDI profile type: Publication
full work available at URL: s00013-009-0080-x
OpenAlex ID: W2000141478
cites work: Q5435174
cites work: Q3999066
cites work: Theta series on the Hecke groups \(G(\sqrt{2})\) and \(G(\sqrt{3})\)
cites work: Q5596313
DOI: 10.1007/S00013-009-0080-X
description: scientific article; zbMATH DE number 5675007