On the Exponential Diophantine Equation (an – 1)(bn – 1) = abx2

Let a and b be distinct fixed positive integers such that min{a, b} > 1. In this paper, we investigate the exponential Diophantine equation (an – 1)(bn – 1) = abx2 in positive integers n and x. By combining methods from the theory of Pell equations, Lucas sequences, and modular arithmetic, we establish that the above equation has no positive integer solutions whenever n is odd. Furthermore, we make a conjecture asserting that the equation possesses a unique solution, namely (a · b, n, x) = (6, 2, 2). As additional evidence supporting this conjecture, we completely solve the case (a, b) = (2, 3), proving that the only solution is (n, x) = (2, 2). We also show that this Diophantine equation admits no solutions for the parameter pairs (a, b) = (2, 7) and (a, b) = (3, 13).