$\displaystyle \frac{1663e^2}{3} \approx 2^{12}$.

showed up in a comment of this mostly-unrelated question.

Numerically, it's not a surprise that $e^2$ is close to a rational number whose numerator and denominator are in this range — similarly good approximations to most numbers can be obtained by truncating the continued fraction at the desired level of accuracy. What's much more of a surprise to me is the appearance of a 12th power in this expression. Is there a good explanation for having such a smooth number here? E.g. see the j-invariant explanation for Ramanujan's observation that

$e^{\pi\sqrt{163}}$

is close to an integer, or the Pisot number explanation for the fact that even powers of the golden ratio are close to integers.

unlikelygiven the size of the numbers involved (taking as a model one in which numbers are selected uniformly at random from some range). $\endgroup$ – Steve Huntsman Nov 19 '11 at 20:26