Incredible Formula - Numberphile
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when discussing the pandigital approximation of e, since it explains the (1+9^(-4^(7*6)))^(3^(2^85)) expression and the pi approximation.
Recommended by 1 people across 1 different months, 2016 to 2019.
What it is
Dr James Grime of Numberphile presents pandigital formulas that use each digit 1 to 9 once, including a huge-exponent expression that approximates e to about 18 trillion trillion digits, plus a shorter pandigital approximation of pi. The video explains why the e formula works and mentions the contest that produced these formulas.
What people said
Recommends Brevan Ellefsen · 2017-02 · Mathematics Stack Exchange · on “What were the mathematical techniques used for estimating euler number e upto 18 trillion trillion digits ??”
Hint: $$\lim_{x \to \infty} (1+1/x)^x = e$$ Hint 2: $$\frac{1}{9^{-4^{6*7}}} = 3^{2^{85}}$$ Also, watch this video by Numberphile on the topic [this video]
Cites AlgorithmsX · 2016-09 · Mathematics Stack Exchange · on “An amazing approximation of $e$”
$$\begin{aligned} (1+9^{-4^{42}})^{3^{2^{85}}} &=(1+9^{-4^{42}})^{3^{2*2^{84}}}\\ &=(1+9^{-4^{42}})^{9^{2^{84}}} \\ &=(1+9^{-4^{42}})^{9^{4^{42}}}\\ &=\Bigl(1+\frac1{9^{4^{42}}}\Bigr)^{9^{4^{42}}}\qquad\text{where }=\left(1+\frac1n\right)^n. \end{aligned}$$ This is just the limit definition of $e$ with a large number as an approximation for $\infty$. Edit: Numberphile [this video] just did a video on this, which also gives a pandigital approximation for $\pi$, but it's only accurate up to ten digits.
Cites migueldva · 2017-11 · Mathematics Stack Exchange · on “On Pandigital numbers”
The other day I ran across a very nice video from Numberphile [this video] in YouTube, which proves that the following formula is approximate to $e$ by $18457734525360901453873570$ digits, which is pretty amazing. \begin{align*} \left(1+9^{-4^{7\cdot 6}}\right)^{3^{2^{85}}} \approx e \end{align*} I got excited about this, and thought about writing an article for my school magazine. Does anyone know any good references (articles, books, videos) that have interesting facts and curiosities concerning pandigital numbers and formulas? Maybe something like the topic addressed in this this question [LINK] is interesting as well, computing algorithms to find pandigital formulas... Thanks in advance, Migu
Cites Sujit Bhattacharyya · 2019-03 · Mathematics Stack Exchange · on “Mathematical coincidences concerning the numbers $\pi$, $e$ and $163$”
Just sharing one of mine. [Sorry: It is only for $e$ and $\pi$ ] $$(1+9^{-4^{6\times 7}})^{3^{2^{85}}}\approx e$$ The left-hand side is equal to first 18,457,734,525,36,901,453,873,570 (18 trillion trillion) digits of $e$ after decimal. See that the left-hand side consists of all the numbers from $1$ to $9$ . A similar thing can be formed for $\pi$ also, $$\pi\approx 2^{5^{.4}}-.6-(\frac{.3^9}{7})^{.8^{.1}}$$ Correct up to 10 digits of $\pi$ . Video_Link [this video]