A class number free criterion for Catalan's conjecture.
type: Item
instance of: scholarly article
title: A class number free criterion for Catalan's conjecture.
zbMATH Open document ID: 1049.11036
DOI: 10.1016/S0022-314X(02)00101-4
published in: Journal of Number Theory
publication date: 2003-05-25T00:00:00Z
review text: Catalan's conjecture [\textit{E. Catalan}, J. Reine Angew. Math. 27, 192 (1844; ERAM 027.0790cj)] predicts that 8 and 9 are the only consecutive integers which are both perfect powers. This conjecture was recently proved by the present author. This paper contains the first part of his contribution to the final solution of this problem. Catalan's conjecture corresponds to the Diophantine equation \(x^p-y^q=1\) where \(p\) and \(q\) are prime numbers. Several arithmetical criteria were obtained by \textit{K. Inkeri} [J. Number Theory 34, 142--152 (1990; Zbl 0699.10029)] and after \textit{M. Mignotte} [C. R. Math. Acad. Sci., Soc. R. Can. 15, 199--200 (1993; Zbl 0802.11010)] and \textit{W. Schwarz} [Acta Arith. 72, 277--279 (1995; Zbl 0837.11014)], but all of them implied some condition on certain class numbers. In this paper the author is able to get rid of these conditions and proves the remarkable fact that if the above equation has a nontrivial solution in rational integers then \[ p^{q-1} \equiv 1 \pmod {q^2} \quad \text{ and} \quad q^{p-1} \equiv 1 \pmod {p^2}. \] The proof follows Inkeri's proof, except for a very ingenious use of Stickelberger's theorem.
Mathematics Subject Classification ID: 11D61
zbMATH DE Number: 1915503
zbMATH Keywords: Catalan's equation
zbMATH Keywords: exponential diophantine equations
Wikidata QID: Q123096814
reviewed by: Maurice Mignotte
MaRDI profile type: Publication
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author: Preda Mihaĭlescu
description: scientific article; zbMATH DE number 1915503