On the number of solutions to systems of Pell equations
type: Item
instance of: scholarly article
title: On the number of solutions to systems of Pell equations
zbMATH Open document ID: 1137.11018
author: Mihai Cipu
author: Maurice Mignotte
published in: Journal of Number Theory
publication date: 2007-09-13T00:00:00Z
review text: Two important results on Diophantine equations are proved in this paper. (1) If \(a>1\), \(b>1\) are distinct integers and \(c,d\) are positive integers, then the system of equations \[ ax^2-cz^2=1, \quad by^2-dz^2=1 \] has at most two solutions in positive integers \(x,y,z\). This result is best possible, since there exist systems as above (actually, infinitely many) for which the system has two solutions. (2) If \(a,b\) are positive integers, then the system of equations \[ x^2-ay^2=1,\quad z^2-bx^2 =1 \] has at most two solutions in positive integers \(x,y,z\). This result, most likely, is not best possible, in the sense that no such system is known with two solutions. The problem of determining an upper bound for the number of solutions to such systems and, more general, to systems with left-hand sides as above and right-hand sides integers not necessarily equal to 1, has attracted the interest of number-theorists for the last 20 years. Much earlier, due to Thue and Siegel it has been known that the number of solutions is finite. The Introduction of the paper contains quite a number of relevant references. Though quite technical, this paper is very well written, so that anybody seriously interested in applications of Baker's method to Diophantine equations will greatly profit from its study. Moreover, I would recommend this paper to graduate students whose research orientation is to classical Diophantine equations as an especially useful reading. A very rough sketch of the proof of the first result follows; the proof of the second result follows similar lines. It is proved first that the number of solutions to the system appearing in (1) is equal to that of a similar system with \(c=a-1\) and \(d=b-1\) (where, of course, \(a\) and \(b\) are, in general, distinct from the \(a\) and \(b\) of the initial system). Next, let \(\alpha=\sqrt{a}+\sqrt{a-1}\), \(\beta=\sqrt{b}+\sqrt{b-1}\). Any solution \(x,y,z\) implies the existence of positive integers \(j,k\) satisfy…
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 11D45
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 5190748
zbMATH Keywords: simultaneous Pell equations
zbMATH Keywords: linear forms in logarithms
reviewed by: Nikos Tzanakis
describes a project that uses: PARI/GP
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.09.016
OpenAlex ID: W2087140361
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DOI: 10.1016/J.JNT.2006.09.016
description: scientific article; zbMATH DE number 5190748