Conway's Tetratri

: 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.Conway and Guy. The Book of Numbers Copernicus. 1995. p.62 It was mentioned in his The 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.Numbers from linear array notation | Steps Toward Infinity!
== 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)
== Computation programs ==
* Mitsuki1729, コンウェイのテトラトリ scratch巨大数選手権コンテスト, scratch. (scratch program for computation of Conway's tetratri)
== Sources ==
== See also ==
*Conway's Tetratet
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