Abstract

In this note, we show that for $ n=4N+3, N \in \mathbb{N}\cup \{0\}$, the exponential Diophantine equation $n^{x}+24^{y}=z^{2}$ has exactly two solutions if $n+1$ or equivalently $N+1$ is an square. When $N+1=m^{2}$, the solutions are given by $(0,1,5)$ and $(1,0,2m).$ Otherwise it has a unique solution $(0,1,5)$ in non-negative integers. Finally, we leave an open problem to explore.

Keywords and Phrases

Catalan's Conjecture solutions, Exponential Diophantine equations, Integer solutions.

A.M.S. subject classification

11D61, 11D72.

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