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What is the smallest integer $n$ for which both $\pi n$ and $e n$ are within $\frac{1}{10,000}$ of an integer?

I have managed to find some where both are within $\frac{1}{1000}$ of an integer, such as $1,892,803$:

$\pi\cdot 1,892,803=5,946,415.99949$ and $e\cdot 1,892,803=5,145,171.99975$.

Can you find any integers $n$ that makes these numbers within $\frac{1}{10,000}$th of an integer? How about $\frac{1}{100,000}$th?

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    May I ask what approach did you follow to find that required integer for 1/1000th case? – 2017-02-18
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    For arbitrary closeness to integers, it looks very deep and I would be surprised if there is an answer . – 2017-02-18
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    At least you can take trivially $n=0$ to satisfy the last two questions. But I think there might be arbitrary many negative numbers $n$ that gives the required precision. – 2017-02-18
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    I'm thinking this should be possible to connect to the theory of diophantine approximations https://en.wikipedia.org/wiki/Diophantine_approximation and continued fractions. Since continued fractions for $\pi$ and $e$ are tabulated that should provide a ready searchspace in which to find integers satifying these conditions in a reasonable time frame. Whether you can get "minimality" that way I'd have to look into. – 2017-02-18

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Here is a systematic way to search for such an integer:

First, look for an integer $r$ such that $r\pi$ is within $1/100,000,000$ of an integer. This search will terminate with $r \le 100,000,000$.

Next, look for an integer $s$ such that $rse$ is within $1/10,000$ of an integer. This search will terminate with $s \le 10,000$.

Now take $n=rs$.

Unfortunately, you asked for the smallest such $n$; and this method is unlikely to find that. But it's a start.