Catalan's equation with a quadratic exponent
type: Item
instance of: scholarly article
zbMATH Open document ID: 1031.11017
author: Tauno Metsänkylä
publication date: 2001-11-13T00:00:00Z
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 1642505
zbMATH Keywords: Catalan equation
MaRDI profile type: Publication
Recommended article: Q4521343
Recommended article: Q4774097
Recommended article: Q4806799
Recommended article: The Catalan equation
Recommended article: Catalan's Equation x p - y q = 1 and Related Congruences
Recommended article: A note on Catalan's equation
Recommended article: Q3520816
title: Catalan's equation with a quadratic exponent
published in: Comptes Rendus Mathématiques de l'Académie des Sciences
review text: The author considers the general case of Catalan's equation, namely \(x^M-y^N=\pm 1\), with \(xy\not=0\) and odd exponents \(1<M<N\). It was proved by the reviewer that \(M\) must be prime and that either \(N\) is prime or \(N=q_1 q_2\) where \(q_1\leq q_2\) are prime numbers. Here the author considers the case \(N=q^2\). In this case he proves the congruences NEWLINE\[NEWLINE x\equiv 0, \quad p^{q-1}\equiv 1 \pmod {q^3}. NEWLINE\]NEWLINE His proof is a generalization of the recent proof of \textit{P. Mihǎilescu} [J. Number Theory 99, 225-231 (2003; Zbl 1049.11036)] for the equation \(x^p-y^q=1\) where \(p\) and \(q\) are prime. More recently, Jon Grantham computed lower bounds on the exponents \(M\) and \(N\) and if these results are compared to upper bounds on \(M\) and \(N\) obtained by the reviewer, it follows that both exponents \(M\) and \(N\) must be prime.
reviewed by: Maurice Mignotte
description: scientific article; zbMATH DE number 1642505