On the Exponential Diophantine Equation 7^x − 5^y = z^2

In this paper, we address the exponential Diophantine equation 7x −5y = z2 , seeking non-negative integer solutions for x, y, and z. Usingmany congruence theorems and Catalan’s conjecture, we prove theexistence 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 ofthis claim by a thorough analysis of computational methods and conceptsfrom number theory. This outcome advances our knowledge of exponentialDiophantine equations and sheds light on how prime numbers andexponentiation interact in these kinds of mathematical investigations.