As we can read in Wolfram Mathworld's article on approximations of $e$, the base of natural logarithm,
An amazing pandigital approximation to e that is correct to $18457734525360901453873570$ decimal digits is given by $$\LARGE \left(1+9^{-4^{6 \cdot 7}}\right)^{3^{2^{85}}}$$ found by R. Sabey in 2004 (Friedman 2004).
The cited paragraph raises two natural questions.
- How was it found? I guess that Sabey hasn't used the trial and error method.
- Using which calculator can I verify its correctness "to $184\ldots570$ decimal digits"?
1 Answer
$\begingroup$$$\begin{aligned} (1+9^{-4^{42}})^{3^{2^{85}}} &=(1+9^{-4^{42}})^{3^{2*2^{84}}}\\ &=(1+9^{-4^{42}})^{9^{2^{84}}} \\ &=(1+9^{-4^{42}})^{9^{4^{42}}}\\ &=\Bigl(1+\frac1{9^{4^{42}}}\Bigr)^{9^{4^{42}}}\qquad\text{where }=\left(1+\frac1n\right)^n. \end{aligned}$$ This is just the limit definition of $e$ with a large number as an approximation for $\infty$.
Edit:Numberphile just did a video on this, which also gives a pandigital approximation for $\pi$, but it's only accurate up to ten digits.
$\endgroup$ 5
