Visualization of penteto using exponentiation

The penteto is equal to s(5,5,4) in strong array notation.[1] The term was coined by wiki user Hyp cos. It can be represented in up-arrow notation as \(5\uparrow\uparrow\uparrow\uparrow5\) (5 hexated to 5).

This number is also equal to s(5,2,5) and s(5,4,1,2).

Etymology

The name of this number is based on the word "penta" and "tetra" (Greek for 5 and 4), meaning a base-5 number, generated using hyper-4 operation based on exponentiation.

Computation

Penteto can be computed in the following process:

  • \(f(1)=5\)
  • \(f(2)=5^{5^{5^{5^5}}}\) with \(f(1)\) 5's, a.k.a. pentbo
  • \(f(3)=5^{5^{\vdots^{5^5}}}\) with \(f(2)\) 5's
  • \(f(4)=5^{5^{\vdots^{5^5}}}\) with \(f(3)\) 5's
  • etc.
  • Penteto equals \(f^{4}(5)\).

Approximations

Notation Approximation
Arrow notation \(5\uparrow\uparrow\uparrow\uparrow5\) (exact)
BEAF \(\{5,5,4\}\) (exact)
Hyper-E notation \(E(5)1\#1\#1\#5\) (exact)
Chained arrow notation \(5\rightarrow5\rightarrow4\) (exact)
Hyperfactorial array notation \(6!3\)
Fast-growing hierarchy \(f_5(4)\)
Hardy hierarchy \(H_{\omega^5}(4)\)
Slow-growing hierarchy \(g_{\varphi(3,0)}(5)\)

Sources

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

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