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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]

Etymology Edit

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 Edit

Boogafive can be computed in the following process:

  • \(f(1)=5\)
  • \(f(2)=5^5=3125\)
  • \(f(3)=5^{5^5}\approx1.911\times10^{2184}\)
  • \(f(4)=5^{5^{5^5}}\)
  • etc.
  • Boogafive is equal to \(f(f(f(f(5))))\).

Approximations Edit

Notation Approximation
Up-arrow notation \(5\uparrow\uparrow\uparrow5\) (exact)
BEAF \(\{5,5,3\}\) (exact)
Hyper-E notation \(E(5)1\#1\#5\) (exact)
Chained arrow notation \(5\rightarrow5\rightarrow3\) (exact)
Hyperfactorial array notation \(6!2\)
Fast-growing hierarchy \(f_4(4)\)
Hardy hierarchy \(H_{\omega^4}(4)\)
Slow-growing hierarchy \(g_{\zeta_0}(5)\)

Sources Edit

  1. Saibian, Sbiis. The Fz, The Fuga & The Megafuga. Retrieved 2017-03-10.
  2. Numbers from linear array notation | Steps Toward Infinity!

See also Edit

Numbers named with the fuga- family of prefixes

Gar-: garone · gartwo · garthree · garfour · garfive · garsix · garseven · gareight · garnine · garten · garhundred · gargoogol · gargoogolplex
Fz-: fzone · fztwo · fzthree · fzfour · fzfive · fzsix · fzseven · fzeight · fznine · fzten · fzeleven · fzhundred · fzgoogol · fzmilliplexion · fzgoogolplex · fzgargoogolplex · fzgargantugoogolplex
Fuga-: fugaone · fugatwo · fugathree · fugafour · fugafive · fugasix · fugaseven · fugaeight · fuganine · fugaten · fugahundred · fugagoogol · fugagoogolplex · fugagargoogolplex · fugagargantugoogolplex
Megafuga-: megafugaone · megafugatwo · megafugathree · megafugafour · megafugafive · megafugasix · megafugaseven · megafugaeight · megafuganine · megafugaten · megafugaeleven · megafugatwelve · megafugathirteen · megafugafourteen · megafugafifteen · megafugasixteen · megafugaseventeen · megafugaeighteen · megafuganineteen · megafugatwenty · megafugahundred · megafugagoogol · megafugagoogolplex · megafugagargoogolplex · megafugagargantugoogolplex
Booga-: boogaone · boogatwo · boogathree · boogafour · boogafive · boogasix · boogaseven · boogaeight · booganine · boogaten · boogahundred · boogagoogol · boogagoogolplex


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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