Some Cases On The Diophantine Equation a^x + b^y = (a + 2m)^z

Abstract

A Diophantine equation is a polynomial equation in several unknowns, where the value of the unknowns are integers. Exponential Diophantine equation is a mathematical equation that involve at least one variable appears as an exponent in the equation. This study investigates specific cases of the exponential Diophantine equation a^x + b^y = (a + 2m)^z, which arises in number theory and has implications in fields such as crypthography and algebraic geometry. We consider the cases where a is a positive integer and x, y, z ≤ 4. The analysis and exploration of solution patterns will be performed using C-language programming implemented in Code::Blocks. This project found that the integral solutions to the equation a^x+b^y = (a+2m)^z are: (a, b, m, x, y, z) = (1, (1+2m)^z-1, m, 1, 1, z), (a, (a+4)^2-a, 2, 1, 1, 2) and for the case of a ∈ Z^+ and (y, z) = (3, 3), there are no positive integral solutions to the equation.