Lower bounds for Catalan's equation
type: Item
instance of: scholarly article
title: Lower bounds for Catalan's equation
zbMATH Open document ID: 0905.11019
DOI: 10.1023/A:1009701725510
author: Maurice Mignotte
author: Yves Roy
published in: The Ramanujan Journal
publication date: 1999-01-21T00:00:00Z
review text: It is conjectured that Catalan's equation \[ x^p - y^q = 1, \quad p \text{ and } q \text{ prime, } \quad x \cdot y \not= 0 \] has only the solution \( (x, y, p, q) = (3, 2, 2, 3). \) Using Baker's results on linear forms in logarithms, \textit{R. Tijdeman} [Acta Arith. 29, 197-209 (1976; Zbl 0286.10013)] showed that there are no solutions if \( \max (p, q) \) is large. By extensive calculations, based on a criterion of Inkeri (in an improved version by Schwarz) on divisibility properties of the relative class number \( h^{-}(K_p) \), where \( K_p \) is a certain subfield of the cyclotomic field \( {\mathbb Q}(e^{2\pi i/p})\), and on an elementary criterion of Mignotte, the authors show: For \(\min(p, q)< 30000\), Catalan's equation has no solution \(\not= (3,2,2,3)\).
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11Y50
zbMATH DE Number: 1185661
zbMATH Keywords: calculations connected with Catalan's equation
zbMATH Keywords: exponential diophantine equations
zbMATH Keywords: computational number theory
zbMATH Keywords: relative class number of cyclotomic fields
zbMATH Keywords: Inkeri's criterion
reviewed by: Wolfgang Schwarz
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1185661