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Visualization of tettro using exponentiation

Tettro is equal to s(4,4,3) in strong array notation.[1] The term was coined by Wikia user Hyp cos. It can be represented in up-arrow notation as \(4\uparrow\uparrow\uparrow4\) (4 pentated to 4). It is approximately 10↑↑10↑↑1010154.

This number is also equal to s(4,2,4) and s(4,3,1,2).

Etymology

The name of this number is based on the word "tetra" and "tri" (Greek for 4 and 3), meaning a base-4 number, generated using hyper-3 operation based on exponentiation.

Computation

Tettro can be computed in the following process:

  • \(f(1)=4\)
  • \(f(2)=4^4=256\)
  • \(f(3)=4^{4^4}\approx1.3408\times10^{154}\)
  • \(f(4)=4^{4^{4^4}}\), a.k.a. tetbo
  • etc.
  • Tettro equals \(f(f(f(4)))\).

Approximations

Notation Approximation
Arrow notation \(4\uparrow\uparrow\uparrow4\) (exact)
BEAF \(\{4,4,3\}\) (exact)
Steinhaus-Moser Notation 4[5]
Hyper-E notation \(\textrm E(4)1\#1\#4\) (exact)
Chained arrow notation \(4\rightarrow4\rightarrow3\) (exact)
Hyperfactorial array notation \(5!2\)
Fast-growing hierarchy \(f_4(3)\)
Hardy hierarchy \(H_{\omega^4}(3)\)
Slow-growing hierarchy \(g_{\zeta_0}(4)\)

See also

Hyp cos' linear strong array notation numbers | exAN numbers
3-entry series
Tribo group: tribo · tetbo · pentbo · hexabo
Trientri group: trientri · tettro · pentro · hextro
Trientet group: triteto · trientet · penteto · hexteto
Trienpent group: tripeno · tetpeno · trienpent · hexpeno
Trienhex group: trihexo · tethexo · penhexo · trienhex

Sources

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