Introduction to BEAF

BEAF, or Bowers Exploding Array Function, is a googological function invented by Jonathan Bowers.

Arrow notation
Before we can understand BEAF, we must understand the extended operators at its core. Arrow notation describes a set of operators invented by Donald Knuth:


 * $$a\uparrow b = a^b$$.
 * $$a\uparrow\uparrow b = \underbrace{a\uparrow a\uparrow\ldots\uparrow a\uparrow a}_b = \underbrace{a^{a^{a^{.^{.^.}}}}}_b$$. Googologists call this tetration; arrows should be solved from right to left.
 * $$a\uparrow\uparrow\uparrow b = \underbrace{a\uparrow\uparrow a\uparrow\uparrow\ldots\uparrow\uparrow a\uparrow\uparrow a}_b$$. Also known as pentation.

In general,
 * $$a\uparrow^{n} b = \underbrace{a\uparrow^{n-1} a\uparrow^{n-1}\ldots\uparrow^{n-1} a\uparrow^{n-1} a}_b$$.

Here are some examples:


 * $$3\uparrow 4 = 3^4 = 81$$
 * $$2\uparrow\uparrow 4 = 2\uparrow 2\uparrow 2\uparrow 2 = 2\uparrow 2\uparrow 4 = 2\uparrow 16 = 65536$$
 * $$4\uparrow\uparrow 4 = 2187879733...5261392896$$; about $$8.0723\cdot 10^{153}$$ digits
 * $$3\uparrow\uparrow\uparrow 3 = 3\uparrow\uparrow 3\uparrow\uparrow 3 = 3\uparrow\uparrow 3^{3^3} = 3\uparrow\uparrow 7625597484987$$