On the Diophantine Equation $(a^n-1)(b^n-1)(c^n-1)=x^2$

In this paper, we study the exponential Diophantine equation $(a^n-1)(b^n-1)(c^n-1)=x^2$ 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 $(a^n-1)(b^n-1)=x^2$ 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 $(2^n-1)(5^n-1)(7^n-1)=x^2$ 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 $(2^n-1)(3^n-1)(5^n-1)=x^2$ has a unique solution $(n, x)=(2, 24)$ under a certain congruence restriction on $n$.