TITLE: Does the sequence $\sin(n!\pi^2)$ converge or diverge?
QUESTION [10 upvotes]: Does the sequence $\sin(n!\pi^2)$ converge or diverge?

REPLY [3 votes]: I devoted a chapter (3) of my thesis on the set $\displaystyle  G = \{x \in \mathbb{R}: \lim_{n \rightarrow \infty} \sin{(n! \pi x) = 0}\}$. It is easy to see that (Euler's constant) $e \in G$. Let alone $\pi$, I do not even know if $e^2 \in G$.