The Diophantine equation \(a^x\pm a^y=z^n\) when \(a\) is any nonnegative integer

Volume 32, Issue 3, pp 213--221 http://dx.doi.org/10.22436/jmcs.032.03.02

Publication Date: September 13, 2023 Submission Date: October 23, 2022 Revision Date: August 01, 2023 Accteptance Date: August 05, 2023

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Authors

K. Laipaporn - School of Science, Walailak University, Nakhon Si Thammarat, 80160, Thailand. S. Wananiyakul - Department of Mathematics and Computer Science, Chulalongkorn University, Bangkok, 10330, Thailand. P. Khachorncharoenkul - School of Science, Walailak University, Nakhon Si Thammarat, 80160, Thailand.

Abstract

In this paper, all solutions of the Diophantine equation \(a^x\pm a^y=z^n\) are investigated when \(a\) is any nonnegative integer and \(n\ge 2\). In particular, if \(p\) is prime and the solutions of \(p^x+p^y=z^n\) exist, then \(p\) is either \(2\) or \(2^n-1\). All proofs in this paper require only elementary number theory.

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Keywords

• Diophantine equation
• Catalan's conjecture
• the fundamental theorem of arithmetic
• Mersenne prime

MSC

•  11D61
•  11D72

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