Non-negative Solutions of the Nonlinear Diophantine Equation (8^n)^x + p^y=z^2  for Some Prime Number p

Boorapa  SINGHA

doi:10.48048/wjst.2021.11719

Authors

Boorapa SINGHA Department of Mathematics and Statistics, Faculty of Science and Technology, Chiang Mai Rajabhat University, Chiangmai 50300, Thailand https://orcid.org/0000-0002-1132-5463

DOI:

https://doi.org/10.48048/wjst.2021.11719

Keywords:

Catalan’s conjecture, Mersenne prime number, Nonlinear Diophantine equation, Non-negative solution

Abstract

In this paper, we explain all non-negative integer solutions for the nonlinear Diophantine equation of type 8^x + p^y= z² when p is an arbitrary odd prime number and incongruent with 1 modulo 8. Then, we apply the result to describe all non-negative integer solutions for the equation (8ⁿ)^x + p^y = z² when n ≥ 2. The results presented in this paper generalize and extend many results announced by other authors.

HIGHLIGHTS

Studying a new series of the equation 8^x + p^y= z² when p is prime which is not congruent to 1 modulo 8
Describing all non-negative integer solutions of the equation (8ⁿ)^x + p^y = z² which is a generalization of the equation 8^x + p^y= z²
The equation 8^x + p^y= z² has at most 3 non-negative integer solutions when p is congruent to 1 modulo 8 and p ≠ 17

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