An extremely good approximation of the Euler number e using only the digits 1-9

Posted: January 15, 2019 in Mathematics

The Euler number, not to confuse with the Euler constant 0.5772156649… , is the irrational number e. It is approximately equal to 2.7182818284…, and it is exactly equal to

\displaystyle \lim_{n\to\infty} \left(1+\frac{1}{n}\right)^{n} \quad\text{ or }\quad \sum_{n=0}^\infty \frac{1}{n!}

Remarkably in around 2004, Richard Sabey gave an extremely accurate approximation to the Euler number,

\displaystyle e \approx (1+9^{-4^{7\cdot6}})^{3^{2^{85}}}

which uses the digits 1 to 9 exactly once and it is accurate to around 1.8 \times 10^{25} digits.

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