Journal of Mathematics and Computer Science, Volume 32, Issue 3, Pages 213-221 , 01/01/2024
The Diophantine equation ax ± ay = zn when a is any nonnegative integer
Abstract
In this paper, all solutions of the Diophantine equation a<sup>x</sup> ± a<sup>y</sup> = z<sup>n</sup> are investigated when a is any nonnegative integer and n ≥ 2. In particular, if p is prime and the solutions of p<sup>x</sup> + p<sup>y</sup> = z<sup>n</sup> exist, then p is either 2 or 2<sup>n</sup> − 1. All proofs in this paper require only elementary number theory.
Document Type
Article
Source Type
Journal
Keywords
Catalan’s conjectureDiophantine equationMersenne primethe fundamental theorem of arithmetic
ASJC Subject Area
Mathematics : Mathematics (all)Mathematics : Computational MathematicsComputer Science : Computer Science ApplicationsEngineering : Computational Mechanics