Exponential Diophantine Equations Involving Opposite Parity Prime

Abstract

Many researchers have been devoted to finding the solutions (x, y,z) in the set of nonnegative integers, of Diophantine equations of the type px +qy = z2, where the values and q are fixed. In this article, we demonstrate that few singular Exponential Diophantine equations E1 : 2x +7y = z2 E2 : 2x +41y = z2, E3 : 2x +43y = z2, E4 : 2x +23y = z2 E5 : 2x +31y = z2 has only a finite number of solutions in N ∪ {0}. The solution sets (x, y,z) of E1,E2,E3,E4 and E5 are {(1,1,3),(3,0,3),(5,2,9)} {(3,0,3),(3,1,7),(7,1,13)},{(3,0,3)},{(3,0,3),(1,1,5)} and respectively