Ballium's number

Ballium's number is a number coined in a parody YouTube video by Meerkats Anonymous, where it is jokingly claimed to be the "largest number ever." According to the video, Ballium's number is exactly

\[(794,843,294,078,147,843,293.7 + 1/30) \cdot e^{\pi^{e^\pi}}\]

The video also goes on to say that the "Microsoft calculator struggles to produce an output, even when set to scientific mode." For obvious reasons, Ballium's number is not actually the largest number ever.

In fact, the number is smaller than a googolplex. This can be shown easily enough, by rounding \(e\) and \(\pi\) up to the next nearest integer and replacing the first component with \(10^{21}\):

\[\text{Ballium's number} < 10^{21} \cdot 3^{4^{3^4}} = 10^{21} \cdot 3^{4^81} = 10^{21} \cdot 3^{2^{162}} = 10^{21} \cdot 3^{10^{162 \log_{10} 2}} < 10^{21} \cdot 3^{10^{162 \cdot 0.4}} = \]

\[10^{21} \cdot 3^{10^{64.8}} < 10^{21} \cdot 10^{10^{64.8}} = 10^{21 + 10^{64.8}} < 10^{10^{64.8} + 10^{64.8}} < 10^{10^{65.8}} < 10^{10^{66}}\]

This is less than \(10^{10^{100}}\), so Ballium's number is smaller than googolplex.

In fact, \(10^{10^{66}}\) is a fairly liberal upper bound. The actual value is closer to \(10^{10^{11}}\). It can be shown that:

\[10^{10^{11}} < \text{Ballium's number} < 10^{10^{12}}\]

Ballium's number contains roughly 138 billion digits before the decimal point, so the process of computing it is impractical, although possible on today's computers.

Ballium's number is a typical example of common attempts to name a very large number. Its form seems to be largely inspired by Skewes' number, but it fails to be as large, mainly because the topmost exponent of the second component is too small.