Approximate Relations between Pi and Euler's Number

Approximate Relations between $\pi$ (pi) and Euler's number $e$

Fanelli's Approximation

This approximation to $\pi$ is accurate to $5$ decimal places:

$\sqrt [9] {10 e^8} = 3 \cdotp 14159 \, 828 \ldots$

Gelfond's Constant minus $\pi$

Gelfond's constant minus $\pi$ is very close to $20$:

$e^\pi - \pi \approx 20$

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

This expression relating $e$ and $\pi$ comes within $10^{-12}$ of an integer:

$e^{\pi \sqrt {163} } \approx 262 \, 537 \, 412 \, 640 \, 768 \, 743 \cdotp 99999 \, 99999 \, 99250 \ldots$