Pandigital approximation to e

I spent some time this week-end trying to find a mathematical puzzle whose solution is 2718, for the “Mathematics on a shelf” competition, and the first trail was to look into properties of Euler’s number e. The following result is not useful in any way, but it is amazing: an approximation of e using all the digits from 1 to 9 exactly once, and which is correct to 18457734525360901453873570 decimal digits (that’s more than 10^{26} digits!):

e \approx \left( 1+9^{-4^{7-6}}\right)^{3^{2^{85}}}.

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3 Responses to “Pandigital approximation to e”

  1. Julyan Says:

    Do you have any clue how one can come to this kind of very precise approximation? A series expansion…?

  2. Nigel Thomas Burin Says:

    There isn’t any luck involved. e is the limit of (1 +t) to the power of 1/t, as t tends to zero. If you look at the given expression, you will see that this is what your approximation is using. t is certainly very small in this case, hence the degree of accuracy of the approximation.
    Nigel Burin ( The Manchester Grammar School)

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