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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the diophantine equation $(x^3-1)/(x-1)=(y^n-1)/(y-1)$
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by Maohua Le PDF
Trans. Amer. Math. Soc. 351 (1999), 1063-1074 Request permission

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
  • Additional Notes: Supported by the National Natural Science Foundation of China and the Guangdong Provincial Natural Science Foundation
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 351 (1999), 1063-1074
  • MSC (1991): Primary 11D61, 11J86
  • DOI: https://doi.org/10.1090/S0002-9947-99-02013-9
  • MathSciNet review: 1443198