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On simultaneous Pell equations and related Thue equations


Authors: Bo He, Ákos Pintér and Alain Togbé
Journal: Proc. Amer. Math. Soc. 143 (2015), 4685-4693
MSC (2010): Primary 11D09, 11D45, 11B37, 11J86
DOI: https://doi.org/10.1090/proc/12608
Published electronically: April 1, 2015
MathSciNet review: 3391027
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we prove that the simultaneous Pell equations

$\displaystyle x^2-(m^2-1)y^2=1,\, z^2-(n^2-1)y^2=1 $

have only a positive integer solution $ (x, y, z) = (m, 1, n)$ if $ m < n \le m+m^{\varepsilon }, 0 < \varepsilon < 1$ and $ m \ge 202304^{\frac {1}{1-\varepsilon }}$. Using a computational reduction method we can omit the lower bound for $ m$ when $ m<n\le m^{\frac {1}{5}}$. Moreover, we apply our main result to a family of Thue equations in two parameters studied by Jadrijević.

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Additional Information

Bo He
Affiliation: Department of Mathematics, Aba Teacher’s College, Wenchuan, Sichuan 623000, People’s Republic of China
Email: bhe@live.cn

Ákos Pintér
Affiliation: Institute of Mathematics, MTA-DE Research Group “Equations, Functions and Curves”, Hungarian Academy of Sciences and University of Debrecen, P.O. Box 12, H-4010 Debrecen, Hungary
Email: apinter@science.unideb.hu

Alain Togbé
Affiliation: Department of Mathematics, Purdue University North Central, 1401 S. U.S. 421, Westville, Indiana 46391
Email: atogbe@pnc.edu

DOI: https://doi.org/10.1090/proc/12608
Received by editor(s): September 4, 2013
Received by editor(s) in revised form: July 31, 2014, and August 13, 2014
Published electronically: April 1, 2015
Additional Notes: The first author was supported by the Natural Science Foundation of China (Grant No. 11301363), the Sichuan provincial scientific research and innovation team in Universities (Grant No. 14TD0040), and the Natural Science Foundation of Education Department of Sichuan Province (Grant No. 13ZA0037 and No. 13ZB0036)
The second author was supported in part by the Hungarian Academy of Sciences, OTKA grants K100339, NK101680, NK104208 and by the European Union and the European Social Fund through project Supercomputer, the national virtual lab (grant no.: TAMOP-4.2.2.C-11/1/KONV-2012-0010)
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2015 American Mathematical Society

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