Return to Question

The numerical coincidence

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

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.

The numerical coincidence

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

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.

The numerical coincidence

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

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?

The numerical coincidence

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

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.

The numerical coincidence

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

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?

The numerical coincidence

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

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?