On the Problem of Solution of Non-Linear (Exponential) Diophantine Equation

Abstract

Diophantine equations have great importance in research and thus among researchers. Algebraic equations with integer coefficients having integer solutions are Diophantine equations. For tackling the Diophantine equations, there is no universal method available. So, researchers are keenly interested in developing new methods for solving these equations. While handling any such equation, three issue arises, that is whether the problem is solvable or not; if solvable, possible number of solutions and lastly to find the complete solutions. Fermat's equation and Pell's equation are most popularly known as Diophantine equations. Diophantine equations are most often used in the field of algebra, coordinate geometry, group theory, linear algebra, trigonometry, cryptography and apart from them, one can even define the number of rational points on circle. In the present manuscript, the authors demonstrated the problem of existence of a solution of a non-linear (exponential) Diophantine equation , where are non-negative integers and are primes such that has the form of a natural number n. After it, authors also discussed some corollaries as special cases of the equation in detail. Results of the present manuscript depict that the equation of the study is not satisfied by the non-negative integer values of the unknowns and . The present methodology of this paper suggests a new way of solving the Diophantine equation especially for academicians, researchers and people interested in the same field.