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Rakporn Dokchan, Nopparat Panngam,
On the Diophantine Equation a^x+(a+5b)^y=z^2.
Int. J. Math. Comput. Sci., 20, no. 1, (2025), 63-66

DOI:

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

Keywords and phrases:

Exponential Diophantine equation.

Abstract:

In this paper, we study the Diophantine equation a^x+(a+5b)^y=z^2 when a\equiv 1\left( \bmod 5 \right) and b is a positive integer. We establish that the equation has no solutions in positive integers x,y and z. We start with the Diophantine equation p^x+(p+5a)^y=z^2 where p and p+5a are both primes and p\equiv 1\left( \bmod 5 \right) and a is a positive integer.