User blog:Undeadlift/Graham's Number using the Supernova Array

Graham's Number is a huge number, and it is has been difficult to write its exact value despite the numerous notations already developed for very large numbers. The Supernova Array offers an alternative method of writing the exact value of Graham's Number with a simple set of rules.

Pertinent Rules

 * Base Function: \(S(a,b) = a\uparrow^{(b)}a\), where \(b\) is the number of \(\uparrow\)'s.
 * \(S(a,b,0) = S(a,b)\)
 * \(S(a,b,c) = S(a,S(a,b),c-1)\)

The other rules are available here.

Writing Graham's Number
We all know that \(3\uparrow \uparrow \uparrow \uparrow 3 = g_1\), \(g_2 = 3 \uparrow^{g_1} 3\) and that Graham's number is \(g_{64}\). Using the Supernova Array:

\(g_1 = S(3,4,0) = S(3,4)\)

\(g_2 = S(3,4,1) = S(3,S(3,4),0) = S(3,S(3,4))\)

\(g_3 = S(3,4,2) = S(3,S(3,4),1) = S(3,S(3,S(3,4)))\)

\(...\)

\(Graham's Number = g_{64} = S(3,4,63)\)