On the Exponential Diophantine Equation 7x − 5y = z2

In this paper, we address the exponential Diophantine equation 7 x − 5 y = z 2 , seeking non-negative integer solutions for x, y, and z. Using many congruence theorems and Catalan's conjecture, we prove the existence of a single solution. Our analysis shows that (x,y,z)=(0,0,0) is the only possible solution to the problem. We prove the validity of this claim by a thorough analysis of computational methods and concepts from number theory. This outcome advances our knowledge of exponential Diophantine equations and sheds light on how prime numbers and exponentiation interact in these kinds of mathematical investigations.