## 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!):

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