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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the diophantine equation
$(x^3-1)/(x-1)=(y^n-1)/(y-1)$


Author: Maohua Le
Journal: Trans. Amer. Math. Soc. 351 (1999), 1063-1074
MSC (1991): Primary 11D61, 11J86
MathSciNet review: 1443198
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Abstract: In this paper we prove that the equation $(x^3-1)/(x-1)=$ $(y^n-1)/(y-1)$, $x,y,n\in\mathbb{N}$, $x>1$, $y>1$, $n>3$, has only the solutions $(x,y,n)=(5,2,5)$ and $(90,2,13)$ with $y$ is a prime power. The proof depends on some new results concerning the upper bounds for the number of solutions of the generalized Ramanujan-Nagell equations.


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Additional Information

Maohua Le
Affiliation: Department of Mathematics, Zhanjiang Teachers College, Postal Code 524048, Zhanjiang, Guangdong, P. R. China

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02013-9
PII: S 0002-9947(99)02013-9
Additional Notes: Supported by the National Natural Science Foundation of China and the Guangdong Provincial Natural Science Foundation
Article copyright: © Copyright 1999 American Mathematical Society