The weighted sum of two S-units being a square
type: Item
instance of: scholarly article
title: The weighted sum of two S-units being a square
zbMATH Open document ID: 0714.11017
DOI: 10.1016/0019-3577(90)90007-A
published in: Indagationes Mathematicae. New Series
publication date: 1990-01-01T00:00:00Z
review text: Let S denote the set of all those integers which are divisible only by the given primes \(p_ 1,...,p_ s\). Let a,b\(\in {\mathbb{Z}}\) such that \(p_ i\nmid ab\) for \(i=1,...,s\) and (a,b) is squarefree. The aim of this paper is to prove that the diophantine equation (1) \(ax+by=z^ 2\) has only finitely many effectively computable solutions in x,y\(\in S\), \(z\in {\mathbb{Z}}\) with (x,y) is squarefree. Actually, this result is a special case of a theorem of \textit{S. V. Kotov} [Acta Arith 31, 219-230 (1976; Zbl 0352.12002)]. But here there are given very precise bounds for linear forms in p-adic logarithm of algebraic numbers, which enables one to compute all the solutions of (1) for given a,b and \(p_ 1,...,p_ s.\) Combining this result with the reduction technique given in the author's book [Algorithms for diophantine equations, CWI Tract 65 (1989; Zbl 0687.10013)] it is proven that (1) has for \(a=1\), \(b=\pm 1\) and \(p_ 1,...,p_ s=2,3,5,7\) exactly 388 solutions.
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 4175019
zbMATH Keywords: linear forms in p-adic logarithm of algebraic numbers
author: Benjamin M. M. de Weger
reviewed by: Attila Pethő
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4175019