Utility of Exponential Diophantine Equation \(3^x + b^y = c^z\)

Abstract

At the current time, we use the Diophantine equation for any polynomial equation with integer coefficients and associated unknowns are taken to be rational integers. This definition is commonly extended to any variety of equations involving integers and where the unknowns are integers. An emblematic example is Fermat's equation xⁿ + bⁿ= cⁿ, where x, b, c and n > 3  are unknown positive integers. We regularly use the terminology “exponential Diophantine equation” when one or more exponents are unknown.
Now consider that for b and c are be a positive integer with fixed coprime here min  c} > 1.In this research article, every positive number solution (x, y, z) of the equation \(3^x + b^y = c^z\) is classified in this section.
Further, we have to prove that, if c = b+ 3, In that case, the equation contains only  positive integer solutions i.e  (x, y, z) = (3, 2, 2), except for (b, x, y, z) = (3, 3, 2, 2) and  (3^r -1, r + 3, 3, 3), Here r is a positive integer number of r \( \geq\) 3 by an elementary approach.