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SOLUTIONS TO A LEBESGUE–NAGELL EQUATION

Part of: Diophantine equations

Published online by Cambridge University Press:  24 May 2021

NGUYEN XUAN THO Open the ORCID record for NGUYEN XUAN THO [Opens in a new window]
NGUYEN XUAN THO*
Affiliation:
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet Road, Hanoi, Vietnam
*
e-mail: [email protected]
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Abstract

We find all integer solutions to the equation $x^2+5^a\cdot 13^b\cdot 17^c=y^n$ with $a,\,b,\,c\geq 0$ , $n\geq 3$ , $x,\,y>0$ and $\gcd (x,\,y)=1$ . Our proof uses a deep result about primitive divisors of Lucas sequences in combination with elementary number theory and computer search.

Keywords

Diophantine equationLebesgue–Nagell equationprimitive divisor theorem

MSC classification

Primary: 11D61: Exponential equations
Secondary: 11D72: Equations in many variables
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 1 , February 2022 , pp. 19 - 30
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

The author is partially supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM) and the Vietnam National Foundation for Science and Technology Development (NAFOSTED) (grant number 101.04-2019.314).

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