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Supawadee Prugsapitak, Phitchayawee Sangjan,

On the Diophantine equation of the form 4^{x'}+(m^2-1)^x=y^2.

Int. J. Math. Comput. Sci., 20, no. 1, (2025), 187-190

DOI:

https://doi.org/10.69793/ijmcs/01.2025/supawadee

Keywords and phrases:

Diophantine Equations, Calatan's Conjecture, Exponential Diophantine Equation.

Abstract:

We provide the sufficient and necessary conditions for the existence of a solution to the Diophantine equation 4^{x'}+(m^2-1)^x=y^2. Moreover, for an even positive integer m, we show that this equation can only be solved if and only if m is a prime power of 2. Furthermore, we provide all of its solutions.