Let us consider a sequence of continuous functions $g_{q}:ℝ^2\to ℝ^2$. Let $(A_{q})_{q\geq 1}$ be a sequence of compact sets in $ℝ^2$. Assuming that each function $g_{q}$ is topologically mixing in $A_{q}$ for all $q\geq 1$, i.e., for every open subsets $U,V$ of $ℝ^2$ such that $U\cap A_{q}$ and $V\cap A_{q}$ are non-empty, there exists $k_0=k_0(q,U,V)$ such that for every $k\geq k_0$ the set $g_{q}^{k}(U)\cap V\cap A_{q}$ is non-empty.

My problem is about finding a common set $A$ in which all the functions $g_{q}$ are topologically mixing in $A$. My approach based on several techniques (symbolic dynamics) leads to this diophantine equation:

$$16(n+1)^2q^8+16(n+1)^2q^6+1=m^2$$

and the main problem is equivalent to the fact that the above diophantine equation has an infinite number of positive integers solutions $q,n,m$. So, the question is how one can prove that the above diophantine equation has an infinite number of positive integers solutions $q,n,m$.