On the positive integral solutions of the Diophantine equation \(x^3+by+1-xyz=0\)
type: Item
instance of: scholarly article
title: On the positive integral solutions of the Diophantine equation \(x^3+by+1-xyz=0\)
zbMATH Open document ID: 1153.11015
author: Florian Luca
author: Alain Togbé
published in: Bulletin of the Malaysian Mathematical Sciences Society. Second Series
publication date: 2008-11-27T00:00:00Z
full work available at URL: https://eudml.org/doc/54919
full work available at URL: v31n2p2.pdf
review text: The authors prove that the title Diophantine equation has at most \(11b+o(b)\) solutions \((x,y,z)\) for fixed \(b\). This is a weak form of a conjecture due to \textit{S. P. Mohanty} and \textit{A. M. S. Ramasamy} [see Bull. Malays. Math. Soc. (2) 7, 23--28 (1984; Zbl 0547.10014)]. Solutions to the title equation fulfill a divisibility condition namely \(xz-b\mid \frac{b^3+z^3}{\gcd(b,z)^2}\) with at most three possible (explicit known) exceptions. By a careful estimation of the sum of divisors \(\sum \tau(z_1^3+b_1^3)\) with \(z=dz_1\), \(b=db_1\) and \(d=\gcd (b,z)\) the authors are able to prove their result. Note \(\tau(x)\) denotes the number of divisors of \(x\).
reviewed by: Volker Ziegler
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11D45
zbMATH DE Number: 5374765
zbMATH Keywords: cubic Diophantine equations
zbMATH Keywords: counting solutions of Diophantine equations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5374765