On Two Classes of Exponential Diophantine Equations

Abstract

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.

(Exponential Diophantine Equation n2􀀀1 )u +n2v = w2;n = 2;3;4;5

“…G. Janaki and C. Saranya (4) established that positive integer solutions exist to the exponential problem using Jarasandha numbers employing the Catalan conjecture. Several exponential Diophantine equations are solved by many authors in (5)(6)(7)(8)(9)(10)(11)(12) .…”

A Paradigm for Two Classes of Simultaneous Exponential Diophantine Equations

“…Simultaneous exponential Diophantine equations are a pair of exponential Diophantine equations whose values satisfy both or all of the equations in the collection at the same time. In (12)(13)(14) , the authors investigate various simultaneous exponential Diophantine equations in the form a 1…”

On the Solution of a Class of Exponential Diophantine Equations

“…Eventually, Preda Mihȃilescu [15] proved the conjecture in 2004. For various values of a and b, the equations a x + b y = z 2 have been vastly studied in non-negative integers, see for example [1,3,4,5,6,12,16,19,21].…”