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 \(n 5\) 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 \(k N\), \((8k+5)^x+24^y=z^2\) has a unique solution \((0,1,5)\) in non-negative integers. We also show that for a perfect square \(8m\), the exponential Diophantine equation \((8m-1)^x+24^y=z^2\), \(m N\) has exactly two non-negative integral solutions \((0,1,5)\) and \((1,0,8m)\). Otherwise, it has a unique solution \((0,1,5)\). Finally, we illustrate our results with some examples and non-examples.