On the Exponential Diophantine Equation (13^2m) + (6r + 1)ⁿ = z²

Abstract

Nowadays, mathematicians are very interested in discovering new and advanced methods for determining the solution of Diophantine equations. Diophantine equations are those equations that have more unknowns than equations. Diophantine equations appear in astronomy, cryptography, abstract algebra, coordinate geometry and trigonometry. Congruence theory plays an important role in finding the solution of some special type Diophantine equations. The absence of any generalized method, which can handle each Diophantine equation, is challenging for researchers. In the present paper, the authors have discussed the existence of the solution of exponential Diophantine equation  (13^2m) + (6r + 1)ⁿ = Z², where m, n, r, z are whole numbers. Results of the present paper show that the exponential Diophantine equation (13^2m) + (6r + 1)ⁿ = Z², where m, n, r, z are whole numbers, has no solution in the whole number.