On the Catalan equation. II
type: Item
instance of: scholarly article
title: On the Catalan equation. II
zbMATH Open document ID: 0797.11036
DOI: 10.1016/0304-3975(94)90075-2
published in: Theoretical Computer Science
publication date: 1994-03-03T00:00:00Z
review text: Consider Catalan's equation \(x^ p - y^ q = 1\), where \(p,q \geq 2\) and \(| x |\), \(| y |>1\). Here \(p\) and \(q\) are chosen to be odd primes with \(p<q\). The author continues his investigations (cf. Sur l'équation de Catalan, [C. R. Acad. Sci., Paris, Sér. I 314, 165-168 (1992; Zbl 0749.11025)]) by introducing a new linear form in logarithms which enables him to improve the general upper bounds on \(p\) and \(q\), in case \(p \equiv 3 \pmod 4\). Moreover, it is shown that Catalan's equation has no solution when \(p = 19\) and that for \(p = 53\), the only possible value for \(q\) is \(q = 4889\).
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 501214
zbMATH Keywords: exponential diophantine equation
zbMATH Keywords: Catalan's equation
zbMATH Keywords: linear form in logarithms
zbMATH Keywords: upper bounds
author: Maurice Mignotte
reviewed by: Roelof J. Stroeker
MaRDI profile type: Publication
full work available at URL: 0304-3975(94)90075-2
OpenAlex ID: W1983026076
cites work: Q3270672
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description: scientific article; zbMATH DE number 501214