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In this paper, we consider the equation $(a^n-2^{m})(b^n-2^{m})=x^2$. Byassuming the abc conjecture is true, in [8], Luca and Walsh gave a theorem,which implies that the above equation has only finitely many solutions $n,x$ ifa and b are different fixed positive integers. We solve the above equation when$m=1$ and $(a,b)=(2,10),(4,100),(10,58),(3,45)$. Moreover, we show that$(a^2-2)(b^2-2)=x^2$ has no solution n,x if 2|n and gcd$(a,b)=1$. We also givea conjecture which says that the equation $(2^2-2)((2P_k)^n-2)=x^2$ has onlythe solution $(n,x)=(2,Q_k)$, where $k>3$ is odd and $P_k,Q_k$ are Pell andPell Lucas numbers, respectively. We also conjecture that if the equation$(a^2-2)(b^2-2)=x^2$ has a solution $n,x$, then $n<7$, where $2<a<b$.

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