On the Diophantine equation \(\frac{x^3-1}{x-1}=\frac{y^n-1}{y-1}\)
type: Item
instance of: scholarly article
title: On the Diophantine equation \(\frac{x^3-1}{x-1}=\frac{y^n-1}{y-1}\)
zbMATH Open document ID: 1063.11009
author: Pingzhi Yuan
published in: Journal of Number Theory
publication date: 2005-05-04T00:00:00Z
review text: The author proves: Theorem. The positive integer solutions \((x,y,n)\) of the equation \[ {x^3-1\over x-1}={y^n-1\over y-1}, \quad \text{with \(n\) odd and \(x\not=y\),} \] are \((x,y,n)=(5,2,5)\) and \((90,2,13)\). The proof uses a previous result of Le and elementary arguments of algebraic number theory for quadratic fields.
reviewed by: Maurice Mignotte
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11J86
Mathematics Subject Classification ID: 11D09
zbMATH DE Number: 2164817
zbMATH Keywords: exponential Diophantine equations
zbMATH Keywords: quadratic equations
MaRDI profile type: Publication
full work available at URL: j.jnt.2004.12.002
OpenAlex ID: W2006559454
cites work: On the equation $a(x^m-1)/(x-1)=b(y^n-1)/(y-1)$.
cites work: Existence of primitive divisors of Lucas and Lehmer numbers (with an appendix by M. Mignotte)
cites work: On the number of solutions of the generalized Ramanujan-Nagell equation
cites work: On the Diophantine equation \(\frac{x^m-1}{x-1}=\frac{y^n-1}{y-1}\).
cites work: EQUATIONS OF THE FORM f(x)=g(y)
cites work: On the diophantine equation $(x^3-1)/(x-1)=(y^n-1)/(y-1)$
cites work: Exceptional solutions of the exponential Diophantine equation \((x^3-1)/(x-1) = (y^n-1)/(y-1)\)
cites work: The Diophantine equation 2𝑥²+1=3ⁿ
cites work: Q3254382
cites work: Q5332261
cites work: The diophantine equation \(x^2+7=2^n\)
cites work: Q5804241
cites work: On an equation of Goormaghtigh
cites work: Integers with identical digits
cites work: Q3695328
cites work: Q3744223
DOI: 10.1016/J.JNT.2004.12.002
description: scientific article; zbMATH DE number 2164817