TITLE: Why is $(\sqrt{2}+\sqrt{3})^{2008}$ so close to an integer?
QUESTION [21 upvotes]: Using 5000-digit precision in PARI/GP, I discovered that the fractional part of $(\sqrt{2}+\sqrt{3})^{2008}$ is extremely small, less than $10^{-999}$. 
Is there a simple explanation for this fact ?
This looks like a Pisot number issue (similar questions have already been studied on MSE, see for example Why is $(2+\sqrt{3})^{50}$ so close to an integer?), but it’s a more complicated situation. 
Related : Show that $(\sqrt{2} + \sqrt{3})^{2009}$ is rounded to an even number.

REPLY [7 votes]: Your intuition is correct:
$$
(\sqrt{2} + \sqrt{3})^{2008} = (5 + 2 \sqrt{6})^{1004}
$$
and $5 + 2\sqrt{6}$ is a Pisot integer. Further, Newton's identities imply that
$$
(5 + 2\sqrt{6})^n + (5 - 2\sqrt{6})^n \in \Bbb{Z}
$$
for every positive integer $n$, thus the distance between $(5 + 2\sqrt{6})^{1004}$ and the closest integer is at most
$$
(5 - 2\sqrt{6})^{1004} \approx 2.6743 \cdot 10^{-1000}
$$