Approximate Relations between Pi and Euler's Number/e to the pi root 163

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

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

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

This sequence is A060295 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Historical Note

The fact that $e^{\pi \sqrt {163} }$ comes so close to an integer was noticed by Alexander Craig Aitken, who considered the property "amusing".

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $163$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $163$