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[]
- ↑ Saibian, Sbiis. The Fz, The Fuga & The Megafuga. Retrieved 2017-03-10.
- ↑ Numbers from linear array notation | Steps Toward Infinity!
- ↑ Hayden's Big Numbers. Retrieved 2022-04-19.
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 · fzthirty · fzhundred · fzthousand · fzmillion · fzgoogol · fzmilliplexion · fzgoogolplex · fzgargoogolplex · fzgiggol
Fuga-: fugaone · fugatwo · fugathree · fugafour · fugafive · fugasix · fugaseven · fugaeight · fuganine · fugaten · fugahundred · fugagoogol · fugagoogolplex · fugagargoogolplex · 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
Tetentri group: duprimitol · duprimitolplex · duprimibol · tetentri
Tetentet group: truprimitol · truprimitolplex · truprimibol · truprimibolplex · tetentet
Tetenpent group: quadprimitol · quadprimitolplex · quadprimibol · quadprimibolplex · tetenpent
Tetenhex group: quinprimitol · quinprimitolplex · quinprimibol · quinprimibolplex · tetenhex
Chainprimol group: chainprimol · chainprimolplex · chainpribol · chainpribolplex · chainduprimol · chainduprimolplex · chaindupribol · chaindupribolplex · chaintruprimol · chaintruprimolplex · chaintrupribol · chaintrupribolplex
Pententri group: duchainol · duchainolplex · duchainbol · duchainbolplex · duchainprimol · duchainprimolplex · duchainpribol · duchainpribolplex · duchainduprimol · duchainduprimolplex · duchaindupribol · pententri
Pententet group: truchainol · truchainolplex · truchainbol · truchainbolplex · truchainprimol · truchainprimolplex · truchainpribol · truchainpribolplex · truchainduprimol · truchainduprimolplex · truchaindupribol · truchaindupribolplex · pententet · pentenpent · pentenhex
Choichainol group: choichainol · choichainolplex · choichainbolplex · choichainpribolplex · choichaindupribolplex · choiduchaindupribolplex
Hexentri group: hexentri · hexentet · hexenpent · hexenhex