TITLE: An inequality for cosine of n
QUESTION [26 upvotes]: Can anyone provide a proof of the following inequality?
If $n$ is a positive integer, $n\geq2$, then $$\cos(n) \leq 1 - 2^{-n}.$$
This is satisfied if $n$ is not within about $2^{-n/2}$ of a multiple of $2\pi$. 
This inequality is sufficient for something else I am trying to prove but I and others have been unable to prove it.

REPLY [32 votes]: It is well known that
$$|\tfrac{n}{k}-\pi|\ge \tfrac1{p(k)}$$
where $p$ is a polynomial of some very finite degree.
From this your estimate follows for all sufficiently large $n$.
The polynomial $p$ can be written explicitly; all you need to find it in the literature and check first few values of $n$ by hands...
P.S. According to K.Mahler 1956, one can take $p(k)=k^{42}$; i.e.; it is sufficient to check all $n\le 1000$.
So, if you trust Martin Brandenburg, your inequality holds for all $n$. ;-)