- Not to be confused with Jonathan Bowers' tetratri.
Conway's Tetratri, also known as Conway's three-three-three-three (or 3-3-3-3) is a large number coined by John Horton Conway.[1] It was mentioned in his Book of Numbers as an example of a number larger than Graham's number using Conway Chained Arrow Notation. Jonathan Bowers once cited it as "the largest number I've seen in the professional literature" on his original 2002 website.
A visual representation of Conway's tetratri, using up-arrow notation
The number is defined as:
\[3 \rightarrow 3\rightarrow 3\rightarrow 3\]
using chained arrows. The number is indeed bigger than Graham's number.
Googology Wiki user Hyp cos calls this number a primibolplex, and it's equal to s(3,3,3,2) in strong array notation.[2]
Proof that Conway's Tetratri > Graham's number
It can be proved without much difficulty that
\[g_{g_{26}} < 3 \rightarrow 3 \rightarrow 3 \rightarrow 3 < g_{g_{27}}\]
Since \(64 < 3^{27} = 3\uparrow\uparrow 3 < 3\uparrow\uparrow\uparrow\uparrow 3 = g_1 << g_{26}\), it follows that:
\[g_{64} << 3 \rightarrow 3 \rightarrow 3 \rightarrow 3\]
Also,
\(3\rightarrow 3\rightarrow 3\rightarrow 3 = 3\rightarrow 3\rightarrow (3\rightarrow 3\rightarrow (3\rightarrow 3)\rightarrow 2)\rightarrow 2 = 3\rightarrow 3\rightarrow (3\rightarrow 3\rightarrow 27\rightarrow 2)\rightarrow 2\)
Since \(G_{64}\) can be approximated as , \(3\rightarrow 3\rightarrow 64\rightarrow 2\),Conway's tetratri is much larger because \(3\rightarrow 3\rightarrow 27\rightarrow 2 >> 64\).
Approximations
(using Cantor normal form's fundamental sequences)
| Notation | Approximation |
|---|---|
| Bowers' Exploding Array Function | \(\{3,3,2,2\}\) |
| Hyper-E notation | \(E2\#\#27\#26\#2\) |
| Hyperfactorial array notation | \((26![2])![2]\) |
| Fast-growing hierarchy | \(f_{\omega+1}(f_{\omega+1}(26))\) |
| Hardy hierarchy | \(H_{\omega^{\omega+1}2}(26)\) |
| Slow-growing hierarchy | \(g_{\Gamma_{\Gamma_0}}(27)\) |
Sources
See also
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