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The Diophantine equation $2 x^2 + 1 = 3^n$


Authors: Ming-Guang Leu and Guan-Wei Li
Journal: Proc. Amer. Math. Soc. 131 (2003), 3643-3645
MSC (2000): Primary 11D61
DOI: https://doi.org/10.1090/S0002-9939-03-07212-5
Published electronically: July 17, 2003
MathSciNet review: 1998169
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Abstract: Let $p$ be a rational prime and $D$ a positive rational integer coprime with $p$. Denote by $N(D, 1,p)$ the number of solutions $(x, n)$ of the equation $D x^2 + 1 = p^n$ in rational integers $x \geq 1$ and $n \geq 1$. In a paper of Le, he claimed that $N(D, 1, p) \leq 2$ without giving a proof. Furthermore, the statement $N(D, 1, p) \leq 2$ has been used by Le, Bugeaud and Shorey in their papers to derive results on certain Diophantine equations. In this paper we point out that the statement $N(D, 1, p) \leq 2$ is incorrect by proving that $N(2, 1, 3)=3$.


References [Enhancements On Off] (What's this?)

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Additional Information

Ming-Guang Leu
Affiliation: Department of Mathematics, National Central University, Chung-Li, Taiwan 32054, Republic of China
Email: mleu@math.ncu.edu.tw

Guan-Wei Li
Affiliation: Department of Mathematics, National Central University, Chung-Li, Taiwan 32054, Republic of China

DOI: https://doi.org/10.1090/S0002-9939-03-07212-5
Received by editor(s): July 2, 2002
Published electronically: July 17, 2003
Additional Notes: The authors research was supported in part by grant NSC 91-2115-M-008-006 of the National Science Council of the Republic of China.
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2003 American Mathematical Society

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