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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 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.

Conway tetratri

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 \(f_{\omega+1}(f_{\omega+1}(26))\)
Hardy hierarchy \(H_{\omega^{\omega+1}2}(26)\)
Slow-growing hierarchy \(g_{\Gamma_{\Gamma_0}}(27)\)

Sources

  1. Conway, John Horton. Book of Numbers
  2. Numbers from linear array notation | Steps Toward Infinity!

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