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