Gelfond's Constant minus Pi
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Theorem
Gelfond's constant minus $\pi$ is very close to $20$:
- $e^\pi - \pi \approx 20$
which is nothing more than an amusing coincidence.
Proof
We have:
- $e^\pi \approx 23 \cdotp 14069 \, 26327 \, 79269 \ldots$
- $\pi \approx 3 \cdotp 14159 \, 26535 \, 89793$
Then:
23.14069 26327 79269 - 3.14159 26535 89793 --------------------- 19.99909 99791 89486 ---------------------
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $23 \cdotp 14069 \, 26327 \, 79269 \, 00572 \, 9086 \ldots$