MegaFavNumbers 262537412680768000
Channel: Richard E Borcherds
Date: 2020-08-22

This video is a contribution to the #MegaFavNumbers project, which 
seems to have overlooked the number 262537412680768000

This number turns up because e to the power of pi times the square root of 163 is about 262537412680768743.99999999999925... with a spectacular sequence of 9s after the decimal point. 

We will discuss how this is related to the elliptic modular function
1/q + 744 + 196884q + 21493760q^2 + ...
and to the monster simple group.

Here are some details about some topics mentioned in the video. 

The high precision calculator free42 is available from https://thomasokken.com/free42/

The video about the monster by 3Blue1Brown is https://www.youtube.com/watch?v=mH0oCDa74tE

The books mentioned at the end are:
"Modular functions and Dirichlet series in number theory" by T. Apostol
"Arithmetic theory of automorphic functions" by G. Shimura
"Sphere packings, lattices, and groups" by J. H. Conway and N. Sloane

Correction: As several alert viewers pointed out, the digit 8 was incorrectly written as 4 in a few places.