A Note on the Exponential Diophantine Equation $(a^m-1)(b^n-1)=x^2$

Abstract:

      Let $a$ and $b$ be fixed positive integers. In this paper, using some elementary methods, we study the diophantine equation $(a^m-1)(b^n-1)=x^2$. For example, we prove that if $a\equiv 2\pmod 6$, $b\equiv 3\pmod{12}$, then $(a^n-1)(b^m-1)=x^2$ has no solutions in positive integers $n,m$ and $x$.