On the Diophantine Equation (aⁿ-1)(bⁿ-1)(cⁿ-1)=x²

In this paper, we study the exponential Diophantine equation (aⁿ-1)(bⁿ-1)(cⁿ-1)=x² in nonnegative integers for certain fixed values of a, b, and c with $ 1$<a<b<c. Our aim is to extend the classical two-factor framework (aⁿ-1)(bⁿ-1)=x² to the corresponding three-factor setting. We first establish a general nonexistence criterion based on $2$-adic valuations and the lifting-the-exponent lemma. As an application, we prove that the equation (2ⁿ-1)(5ⁿ-1)(7ⁿ-1)=x² has no positive integer solutions. Moreover, we derive a more general result covering a family of triples $(a,b,c)$ under suitable parity conditions. Furthermore, we prove that the equation (2ⁿ-1)(3ⁿ-1)(5ⁿ-1)=x² has a unique solution $(n, x)=(2, 24)$ under a certain congruence restriction on n.