3 rewrite to avoid inane answers

$\displaystyle 1663 \approx frac{1663e^2}{3} \frac{3\times 2^{12}}{e^2}$approx 2^{12}$. showed up in a comment of this mostly-unrelated question. Numerically, it's not a surprise that$e^2$is close to a rational number whose numerator and denominator are in this range similarly good approximations to most numbers can be obtained by truncating the continued fraction at the desired level of accuracy. What's much more of a surprise to me is the appearance of a 12th power in this expression. Is there a good explanation for this, e.ghaving such a smooth number here? E.g. like see the j-invariant explanation for Ramanujan's observation that$e^{\pi\sqrt{163}}$is close to an integer, or like the Pisot number explanation for the fact that even powers of the golden ratio are close to integers?. 2 added 3 characters in body$\displaystyle 1663 \approx \frac{3\times 2^{12}}{e^2}$. showed up in a comment of this mostly-unrelated question. Is there a good explanation for this, e.g. like the j-invariant explanation for Ramanujan's observation that$e^{\pi\sqrt{163}}$is close to an integer, or like the fact that even powers of the golden ratio are close to integers? 1 # Numerical coincidence involving the number 1663$\displaystyle 1663 \approx \frac{3\times 2^{12}}{e^2}$. showed up in a comment of this mostly-unrelated question. Is there a good explanation for this, e.g. like the j-invariant explanation for Ramanujan's observation that$e^{\pi\sqrt{163}}\$

is close to an integer, or like the fact that powers of the golden ratio are close to integers?