  In 2002, F. Luca and G. Walsh solved the Diophantine equation in the title
for all pairs (a,b) such that 1<a<b<101 with some exceptions. There are sixty
nine exceptions. In this paper, we give some new results concerning the
equation in the title. It is proved that the equation (a^n-1)(b^n-1)=x^2 has no
solutions if a,b have opposite parity and n>4 with 2|n. Also, we solved
(a^n-1)(b^n-1)=x^2 for the pairs
(a,b)=(2,50),(4,49),(12,45),(13,76),(20,77),(28,49), and (45,100). Lastly, we
show that when b is even, the equation (a^n-1)(b^(2n)a^n-1)=x^2 has no
solutions n,x.
