On Two Classes of Exponential Diophantine Equations - Borah, Padma Bhushan; Dutta, Mridul (Communications in Mathematics and Applications, 2022)
In this paper, we study on the exponential Diophantine equations: \(n^{x}+24^{y}=z^{2}\), for \(n \equiv 5\) or  7 (mod 8). We show that \(5^{x}+24^{y}=z^{2}\) has a unique positive integral solution \((2,1,7)\). Further, we show that for \(k \in \mathbb{N}\), \((8k+5)^{x}+24^{y}=z^{2}\) has a unique solution \((0,1,5)\) in non-negative integers. We also show that for a perfect square \(8m\), the exponential Diophantine equation \((8m-1)^{x}+24^{y}=z^{2}\), \(m \in \mathbb{N}\) has exactly two non-negative integral solutions \((0,1,5)\) and \((1,0,\sqrt{8m})\). Otherwise, it has a unique solution \((0,1,5)\). Finally, we illustrate our results with some examples and non-examples.
Keywords: Catalan's conjecture; Exponential Diophantine equations; Integer solutions