$\frac{b^3-1}{ab-1}=x$ Firstly notice that if $(a,b)$ is a solution, then $(b,a)$ is also a solution. So, we may assume wlog $a \ge b$ . If $b=1$, $x=0$ . So, we got a solution $(a,1)$ where $a>1$ . Now consider the case $b \ge 2$ . $x>0$. Notice that $x \equiv 1 (mod b&...