Googology Wiki
Advertisement
Googology Wiki

The hyperlicious function is defined as

  • \(h_x(a,b) = a\uparrow^{x-2} b \),
  • \(h_x(a,b,c,\ldots,m,n,1) = h_x(a,b,c,\ldots,m,n)\), and
  • \(h_x(a,b,c,\ldots,m,n) = h_{h_x(a,b,c,\ldots,m - 1)}(a,b,c,\ldots,m)\).[1][dead link]

Up to \(h_x(a,b)\)

Examples

  • \(h_3(2,6) = 2 \uparrow 6\)
  • \(h_3(10,100) = 10 \uparrow 100\) (googol)
  • \(h_4(10,100) = 10 \uparrow\uparrow 100\) (giggol)
  • \(h_5(10,100) = 10 \uparrow\uparrow\uparrow 100\) (gaggol)
  • \(h_4(2,6,2) = h_{h_4(2,5)}(2,6) = h_{2\uparrow^6 5}(2,6) = 2 \uparrow^{2 \uparrow^{6} 5-2} 6\)

Growth rate

\(h(x) = h_x(x,x...x,x)\) with x x's in the array is approximately \(f_{\omega+1}(x)\) in the fast growing hierarchy, which is closely related to the expansion function ({x,x,1,2}). The two-entry hyperlicious function (\(h_x(a,b)\)) eventually dominates any hyper-operator, such as tetration, pentation, or even beyond centation.

Sources

See also

Advertisement