On the Exponential Diophantine Equation $(a^{n}-2)(b^{n}-2)=x^{2}$

In this paper, we deal with the equation $(a^{n}-2)(b^{n}-2)=x^{2}$, $2\leq a<b$, and $a,b,x,n\in\mathbb{N}$. We solve this equation for $(a,b)\in\{(2,10),(4,100),(10,58),(3,45)\}$. Moreover, we show that $(a^{n}-2)(b^{n}-2)=x^{2}$ has no solution $n,x$ if $2|n$ and $\gcd(a,b)=1$. We also give a conjecture which says that the equation $(2^{n}-2)((2P_{k})^{n}-2)=x^{2}$ has only the solution $(n,x)=(2,Q_{k})$, where $k>3$ is odd and $P_{k},Q_{k}$ are the Pell and Pell Lucas numbers, respectively. We also conjecture that if the equation $(a^{n}-2)(b^{n}-2)=x^{2}$ has a solution $n,x,$ then $n\leq6$.