TITLE: Are sines of primes dense in $[-1,1]?$
QUESTION [32 upvotes]: Let $P$ be the set of all prime numbers. Is $\sin(P)$ dense is $[-1,1]?$ How could we approach such a problem?

REPLY [18 votes]: According to the Wikipedia article about the discrepancy of a sequence:

The sequence of all multiples of an irrational $\alpha$ by successive prime
  numbers, $2 \alpha$, $3 \alpha$, $5 \alpha$, $7 \alpha$, $11 \alpha$, ... is
  equidistributed modulo 1. This is a famous theorem of analytic number
  theory, proved by I. M. Vinogradov in 1935.

With $\alpha = \frac{1}{2 \pi}$, this implies that $P$ is equidistributed modulo $2 \pi$.   Using this, and the continuity of the sine function, I think it is straightforward to show that $\sin(P)$ is dense in $[-1,1]$ (although not equidistributed).