Divisibility of ceiling function of surd Show that:
The integer next greater than  $(\sqrt{7}+\sqrt{3})^{2n}$ is divisible by $4^n$

Since
$$
(\sqrt7+\sqrt3)^2+(\sqrt7-\sqrt3)^2=20\tag{1}
$$
and
$$
(\sqrt7+\sqrt3)^2(\sqrt7-\sqrt3)^2=16\tag{2}
$$
$(\sqrt7+\sqrt3)^2$ and $(\sqrt7-\sqrt3)^2$ satisfy the equation
$$
x^2-20x+16=0\tag{3}
$$
The recursion
$$
a_n=20a_{n-1}-16a_{n-2}\tag{4}
$$
where $a_0=2$ and $a_1=20$ is therefore satisfied by
$$
a_n=(\sqrt7+\sqrt3)^{2n}+(\sqrt7-\sqrt3)^{2n}\tag{5}
$$
Since $(\sqrt7-\sqrt3)^2\lt1$, we have
$$
a_n=\left\lceil(\sqrt7+\sqrt3)^{2n}\right\rceil\tag{6}
$$
Divide $(4)$ by $4^n$ to get
$$
\frac{a_n}{4^n}=5\frac{a_{n-1}}{4^{n-1}}-\frac{a_{n-2}}{4^{n-2}}\tag{7}
$$
Since $\dfrac{a_0}{4^0}=2$ and $\dfrac{a_1}{4^1}=5$, what does $(7)$ tell you about $a_n$? Combine with $(6)$.
