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. Famous functions such as the fast-growing hierarchy and chained arrow notation only come up with approximations, while the Graham Array Notation and G function give the exact value using complex definitions. The Supernova Array offers an alternative method of writing 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 \(g_{64}\) or \(3\uparrow^{(g_{63})}3\). Using the Supernova Array:

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

\(g_2 = S(3,4,1) = S(3,S(3,4),0) = S(3,S(3,4)) = S(3,g_1) = 3\uparrow^{(g_1)}3\)

\(g_3 = S(3,4,2) = S(3,S(3,4),1) = S(3,S(3,S(3,4))) = S(3,g_2) = 3\uparrow^{(g_2)}3\)


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

If you'll notice, \(g_{n+1} = S(3,4,n) = 3\uparrow^{(g_n)}3\).

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