Why is $ (2+\sqrt{3})^n+(2-\sqrt{3})^n$ an integer? Answers to limit $\lim_{n\to ∞}\sin(\pi(2+\sqrt3)^n)$ start by saying that $ (2+\sqrt{3})^n+(2-\sqrt{3})^n $ is an integer, but how can one see that is true? 
Update: I was hoping there is something more than binomial formula for cases like $ (a+\sqrt[m]{b})^n+(a-\sqrt[m]{b})^n $ to be an integer

Let $x \in \Bbb Q(\sqrt 3)$ be your number. Since $x$ is fixed by any $\sigma \in \text{Gal}(\Bbb Q(\sqrt 3)/\Bbb Q)$, we know by Galois theory that $x$ is rational. As it is an algebraic integer, it is an integer.

This can be used to show that
$$y=(3-2\cos(2\pi/7))^4+(3-2\cos(4\pi/7))^4+(3-2\cos(6\pi/7))^4$$ is also an integer ($y=682$).
(The minimal polynomial of $2\cos(2\pi/7)$ over $\Bbb Q$ is $x^3+x^2-2 x-1$; the $3$ conjugates of
$2\cos(2\pi/7)=\zeta_7+\overline{\zeta_7}=\zeta_7+\zeta_7^{-1},$
 with $\zeta_7=e^{2\pi i /7}$, are of the form $\zeta_7^k+\zeta_7^{-k}=2\cos(2k\pi/7)$).
