(Exponential Diophantine Equation n21 )u +n2v = w2;n = 2;3;4;5

Objectives: Diophantine research focuses on various ways to tackle multivariable and multidegree Diophantine problems. A Diophantine equation is a polynomial equation with only integer solutions. The objective of this manuscript is to find the solutions to a few exponential Diophantine equations and . Also generalize the Exponential equation , and of the form and explore that it has at least one solution as . Methods: Diophantine equations may have finite, infinite or no solutions in integers. There is no universal method for finding solutions to Diophantine equations. The particular type of Exponential Diophantine equation is analysed and generalised by the method of Catalan\'s conjecture. Findings: Exponential Diophantine equations and has only a finite number of solutions in (Whole numbers). The solution sets of , and are, respectively. Novelty: In this analysis, the particular type of Exponential Diophantine equation is analysed using elementary mathematics concepts instead of higher mathematics also generalize the Exponential equation , and of the form and explore that it has at least one solution as . 2020 Mathematical Subject Classification: 11D61. Keywords: Catalan’s conjecture; Diophantine equation; Exponential Diophantine equation; Integral solutions; Non-negative integer solution