Exponential diophantine equations with four terms
type: Item
instance of: scholarly article
title: Exponential diophantine equations with four terms
zbMATH Open document ID: 0765.11018
DOI: 10.1016/0019-3577(92)90045-M
author: Deze Mo
author: R. Tijdeman
published in: Indagationes Mathematicae. New Series
publication date: 1992-09-27T00:00:00Z
review text: This article gives some examples how to make exponential diophantine equations more practical. The authors take the large exponential bounds for solutions given by Baker's method to computational available bounds. Let \(p\) and \(q\) be distinct primes less than 200. The main theorems are: (1) Every solution of the equation \(p^ x q^ y\pm p^ z \pm q^ w \pm 1=0\) with \(zw>0\) satisfies \(\max(p^ x,q^ y,p^ z,q^ w)\leq 2^{15}\). (2) Every solution of the equation \(p^ x\pm q^ y\pm p^ z\pm q^ w=0\) subject to \(x\geq z\), \(y\geq w\), \(p^ x>q^ y\), \(xy>zw\) satisfies \(\max(p^ x,q^ y,p^ z,q^ w)\leq 2^{15}\). (3) Let \(n\) be a positive integer with \(n\leq 5000\). All solutions of \(p^ x-q^ y=n\) satisfy \(\max(p^ x,q^ y)\leq 29^ 4=707281\). The only solutions with \(p^ x>199^ 2\) are given by \(29^ 4-89^ 3=2312\), \(5^ 8-73^ 3=1608\), \(2^{17}-19^ 4=751\), \(17^ 4-43^ 3=4041\), \(43^ 3-5^ 7=1382\), \(41^ 3-2^{16}=3385\).
Mathematics Subject Classification ID: 11D61
zbMATH DE Number: 57231
zbMATH Keywords: exponential diophantine equations
zbMATH Keywords: computational available bounds
reviewed by: Bernd Richter
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 57231