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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. It is slightly smaller and comparable to Saibian's graatagolda-sudex.

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
BEAF \(\{3,3,2,2\}\)
Hyper-E notation \(E2\#\#27\#26\#2\)
Hyperfactorial array notation \((26![2])![2]\)
Strong array notation s(3,3,3,2)
Fast-growing hierarchy \(f_{\omega+1}(f_{\omega+1}(26))\)
Hardy hierarchy \(H_{\omega^{\omega+1}2}(26)\)
Slow-growing hierarchy \(g_{\Gamma_{\Gamma_0}}(27)\)

Computation programs

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

Original numbers, functions, notations, and notions

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's psi function
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4 original idea
By ふぃっしゅ (Fish): Fish numbers (Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7 · S map · SS map · s(n) map · m(n) map · m(m,n) map) · Bashicu matrix system 1/2/3/4 formalisation · TR function (I0 function)
By Gaoji: Weak Buchholz's function
By じぇいそん (Jason): Irrational arrow notation · δOCF · δφ · ε function
By 甘露東風 (Kanrokoti): KumaKuma ψ function
By koteitan: Bashicu matrix system 2.3
By mrna: 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ
By Naruyoko Naruyo: Y sequence formalisation · ω-Y sequence formalisation
By Nayuta Ito: N primitive · Flan numbers (Flan number 1 · Flan number 2 · Flan number 3 · Flan number 4 version 3 · Flan number 5 version 3) · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4
By Okkuu: Extended Weak Buchholz's function
By p進大好きbot: Ordinal notation associated to Extended Weak Buchholz's function · Ordinal notation associated to Extended Buchholz's function · Naruyoko is the great · Large Number Garden Number
By たろう (Taro): Taro's multivariable Ackermann function
By ゆきと (Yukito): Hyper primitive sequence system · Y sequence original idea · YY sequence · Y function · ω-Y sequence original idea


Methodology

By バシク (BashicuHyudora): Bashicu matrix system as a notation template
By じぇいそん (Jason): Shifting definition
By mrna: Side nesting
By Nayuta Ito and ゆきと (Yukito): Difference sequence system



Proofs, translation maps for analysis schema, and other mathematical contributions

By ふぃっしゅ (Fish): Translation map for primitive sequence system and Cantor normal form
By Kihara: Proof of an estimation of TREE sequence · Proof of the incomparability of Busy Beaver function and FGH associated to Kleene's \(\mathcal{O}\)
By koteitan: Translation map for primitive sequence system and Cantor normal form
By Naruyoko Naruyo: Translation map for Extended Weak Buchholz's function and Extended Buchholz's function
By p進大好きbot: Proof of the termination of Hyper primitive sequence system · Proof of the termination of Pair sequence number · Proof of the termination of segements of TR function in the base theory under the assumption of the \(\Sigma_1\)-soundness and the pointwise well-definedness of \(\textrm{TR}(T,n)\) for the case where \(T\) is the formalisation of the base theory


Entertainments

By 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
By koteitan: Dancing video of a Gijinka of Fukashigi · Dancing video of a Gijinka of 久界 · Storyteller's theotre video reading Large Number Garden Number aloud


See also: Template:Googology in Asia
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