Ballium's number

Ballium's Number was part of a video spoof about the "largest number". In the video it is claimed that scientist Samuel Ballium had discovered the highest number one night while reading shakespeare. According to the video the "highest number" is:

(794,843,294,078,147,843,293 .7+1/30)*e^pi^e^pi

The video also goes on to say that the "microsoft calculator struggles to produce an output even when set to scientific mode".

Ironically, despite the numbers claim to being the "highest number", it turns out that despite the stack of exponents this number is even smaller than a googolplex. This can be shown easily enough, by rounding e and pi up to the next nearest integer and replacing 794,843,294,078,147,843,,293.7+1/30 with E21.

Ballium's Number < (E21)*3^4^3^4 = E21*3^4^81 = E21*3^2^162 = E21*3^10^(162log2) < E21*3^10^(162*0.4) =

E21*3^10^64.8 < E21*10^10^64.8 = 10^(21 + 10^64.8) < 10^(10^64.8 + 10^64.8) < 10^10^65.8 < 10^10^66 =

E66#2 < E100#2 = googolplex.

In fact, E66#2 is a rather large overestimate. The actual value is much closer to E11#2. It can be shown that:

E11#2 < Ballium's Number < E12#2

Ballium's Number contains roughly 138 billion digits before the decimal point, so it's beyond what's practical to compute, even to the nearest integer value, although it could be done in theory at least.

Ballium's Number is a typical example of common attempts to name a very large number. It's form seems to be largely inspired by Skewes' Number, but it fails to be as large, mainly because it's top most exponent is too small (Skewes' leading exponent was 79). Skewes' Number however, hasn't been competitive since 1933, and googologist's study much much larger numbers than these. Consequently googologist's would be out of business today if this was the "highest number".

Sources

https://sites.google.com/site/largenumbers/home/a-1/numbers_page

http://www.youtube.com/watch?v=4ypU7YG_bBk