As usual I expect to be critisised for "duplicating"
[this question](https://mathoverflow.net/questions/27931/). But I do not! As Gjergji immediately
notified, that question was from numerology. The one I ask you here
(after putting it in my [response](https://mathoverflow.net/questions/27931/why-is-163-ln163-a-near-integer/28002#28002)) is a mathematics
question motivated by Kevin's (O'Bryant) comment to the earlier post.


**Problem.**
For any $\epsilon>0$, there exists an $n$ such that 
$\|n/\log(n)\|<\epsilon$ where $\|\ \cdot\ \|$ denotes the
distance to the nearest integer.


In spite of the simple formulation, it is likely that
the diophantine problem is open. I wonder whether it follows
from some known conjectures (for example, Schanuel's conjecture).



When is $n/\ln(n)$ close to an integer?
If $f(x)=\frac{x}{\log x}$, then $f'(x)=\frac{1}{\log x} - \frac{1}{(\log x)^2}$, which tends to zero as $x\rightarrow \infty$. Choose some large real number $x$ for which $f(x)$ is integral. Then the value of $f$ on any integer near $x$ must be very close to integral.


