The Diophantine equation \((a^n-1)(b^n-1)=x^2\) (Q1872897)

In earlier papers, \textit{L. Szalay} [Publ. Math. 57, 1-9 (2000; Zbl 0961.11013)] and \textit{L. Szalay} and \textit{L. Hajdu} [Period. Math. Hung. 40, 141-145 (2000; Zbl 0973.11015)] completely solved the title equation when \(\{a,b\}=\{2,3\},\{2,2^k\}\), and \(\{a,b\}=\{2,6\},\{a,a^k\}\), respectively. In the present paper the author considerably generalizes these theorems. Among other results, in the case \(a^l=b^k\) all solutions of the title equation are described. The author completely solves the title equation when \(2\leq a<b\leq 12\) and \(\{a,b\}\neq\{2,10\}\), as well. Some interesting conjectures are also formulated.