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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.

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 $$g_{\Gamma_{\Gamma_0}}(27)$$

## Sources

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
5-entry series
6-entry series

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