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 H. 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

Notation Approximation
BEAF \(\{3,3,2,2\}\)
Hyper-E notation \(E2\#\#27\#26\#2\)
Chained arrow notation \(3\rightarrow3\rightarrow3\rightarrow3\) (exact)
Hyperfactorial array notation \((26![2])![2]\)
Fast-growing hierarchy (using CNF's fundamental sequences) \(f_{\omega+1}(f_{\omega+1}(26))\)
Hardy hierarchy \(H_{\omega^{\omega+1}2}(26)\)
Slow-growing hierarchy \(g_{\Gamma_{\Gamma_0}}(27)\)

Sources

See also

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

 (using CNF's fundamental sequences)

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