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Pentro

Visualization of boogafive using exponentiation

Boogafive is equal to \(5\uparrow\uparrow\uparrow5\), or 5 pentated to 5.[1]

Googology Wiki user Hyp cos calls this number pentro, and it's equal to s(5,5,3) = s(5,2,4) = s(5,3,1,2) in strong array notation.[2]

Googology Wiki user HaydenTheGoogologist2009 calls this number Hayden's Tripent, and it's equal to 5(5)5 in Hayden's Array Notation.[3]

Etymology[]

The name "boogafive" is formed by adding the booga- prefix to 5.

The name "pentro" is based on the word "penta" and "tri" (Greek for 5 and 3), meaning a base-5 number, generated using hyper-3 operation based on exponentiation.

Computation[]

Boogafive can be computed in the following process:

  • \(f(1)=5\)
  • \(f(2)=5^5=3,125\)
  • \(f(3)=5^{5^5}\approx1.911\times10^{2,184}\)
  • \(f(4)=5^{5^{5^5}}\)
  • etc.
  • Boogafive is equal to \(f(f(f(f(5))))\).

Approximations[]

Notation Approximation
Arrow notation \(5\uparrow\uparrow\uparrow5\) (exact)
Bowers' Exploding Array Function \(\{5,5,3\}\) (exact)
Bird's array notation \(\{5,5,3\}\) (exact)
Chained arrow notation \(5\rightarrow5\rightarrow3\) (exact)
Hyper-E notation \(E(5)1\#1\#5\) (exact), \(E2184.1257220888506\#3\#4\)
Hyperfactorial array notation \(6!2\)
Steinhaus-Moser Notation 5[5]
Strong array notation s(5, 5, 3) (exact)
X-Sequence Hyper-Exponential Notation 5{3}3, 5{X}3 (both are exact)
Fast-growing hierarchy \(f_4(4)\)
Hardy hierarchy \(H_{\omega^4}(4)\)
Slow-growing hierarchy \(g_{\zeta_0}(5)\)

Sources[]

See also[]

Gar-: garone · gartwo · garthree · garfour · garfive · garsix · garseven · gareight · garnine · garten · garhundred · garmillion · gargoogol · gareceton · gartrialogue · gargoogolplex · gargiggol
Fz-: fzone · fztwo · fzthree · fzfour · fzfive · fzsix · fzseven · fzeight · fznine · fzten · fztwenty · fzhundred · fzthousand · fzmillion · fzgoogol · fzmilliplexion · fzgoogolplex · fzgargoogolplex · fzgargantugoogolplex · fzgiggol
Fuga-: fugaone · fugatwo · fugathree · fugafour · fugafive · fugasix · fugaseven · fugaeight · fuganine · fugaten · fugahundred · fugagoogol · fugagoogolplex · fugagargoogolplex · fugagargantugoogolplex · fugagiggol
Megafuga-: megafugaone · megafugatwo · megafugathree · megafugafour · megafugafive · megafugasix · megafugaseven · megafugaeight · megafuganine · megafugaten · megafugahundred · megafugagoogol · megafugagoogolplex · megafugagargoogolplex · megafugagargantugoogolplex · megafugagrangol
Extensions:
Booga-:: boogaone · boogatwo · boogathree · boogafour · boogafive · boogasix · boogaten · boogahundred · boogagoogol · boogagoogolplex
Gag-:: gagone · gagtwo · gagthree · gagfour · gagfive · gagsix · gagseven · gageight · gagnine · gagten · gaggoogol · gaggoogolplex

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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