On the diophantine equation \(ap^ x+bq^ y=c+dp^ zq^ w\)
type: Item
instance of: scholarly article
title: On the diophantine equation \(ap^ x+bq^ y=c+dp^ zq^ w\)
zbMATH Open document ID: 0703.11017
DOI: 10.1016/0022-314X(90)90112-5
author: Christopher M. Skinner
published in: Journal of Number Theory
publication date: 1990-01-01T00:00:00Z
review text: It is shown that the title equation (an exponential diophantine equation in x, y, z, and w, where p, q, a, b, c, d are fixed positive integers with \((p,q)=1)\) can be essentially solved, by proving the existence of effectively computable upper bounds on the solutions. Use is made of theorems of \textit{J.-H. Evertse} [Compos. Math. 53, 225-244 (1984; Zbl 0547.10008), Corollary 1, p. 225], \textit{A. J. van der Poorten} [Transcend. Theory, Adv. Appl., Proc. Conf., Cambridge 1976, 29-57 (1977; Zbl 0367.10034), Theorem 1] and \textit{M. Waldschmidt} [Acta Arith. 37, 257-283 (1980; Zbl 0357.10017), Main result]. The author claims that all constants involved can be explicitly computed in terms of the parameters. Moreover, the large bounds may be reduced so as to allow the practical solution of the equation. Demonstrations are given by the special equations \((p,q,a,b,c,d)=(2,3,1,1,1,1)\) and (2,3,5,7,11,1).
Mathematics Subject Classification ID: 11D61
zbMATH DE Number: 4152464
zbMATH Keywords: exponential diophantine equation
zbMATH Keywords: existence of effectively computable upper bounds
reviewed by: Roelof J. Stroeker
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4152464