Conway's Tetratet

Conway's Tetratet, or Conway's four-four-four-four, is a number mentioned by John Horton Conway in the Book of Numbers.Conway, John Horton. Book of Numbers It is the largest explicitly defined finite number in the entire book. It is also the 4th Conway number.
The number is defined as:
\[4\rightarrow 4\rightarrow 4\rightarrow 4\]
in chained arrow notation. It is the smallest nth Conway number to be larger than the nth Ackermann number.
Using Bird's Proof, it can be shown that \(\{4,5,3,2\} \geq 4 \rightarrow 4 \rightarrow 4 \rightarrow 4\) in BEAF.
Conway's Tetratet is the output of the CG function, by putting the input with 4, CG(4).
\(4\rightarrow 4\rightarrow 4\rightarrow 4 = 4\rightarrow 4\rightarrow (4\rightarrow 4\rightarrow (4\rightarrow 4\rightarrow (4\rightarrow 4)\rightarrow 3)\rightarrow 3)\rightarrow 3 = 4\rightarrow 4\rightarrow (4\rightarrow 4\rightarrow (4\rightarrow 4\rightarrow 256\rightarrow 3)\rightarrow 3)\rightarrow 3\)
== Approximations ==
Notation 	Approximation
BEAF 	\(\{4,5,3,2\}\)
Extended Hyper-E notation 	\(E4\#\#4\#\#4\)
Fast-growing hierarchy (using CNF's fundamental sequences)	\(f_{\omega+2}(f_{\omega+2}(f_{\omega+2}(255)))\)
Hardy hierarchy 	\(H_{\omega^{\omega+2}3}(255)\)
Slow-growing hierarchy 	\(g_{\varphi(1,1,0)}(255)\)
Hyperfactorial array notation 	\(4![1,2] = 4![4]\)
Notation array 	\((4\{3,4\}4)\)
== See also ==
*Conway's Tetratri
== Sources ==
Category:Numbers
Category:Eponymous numbers
Category:Greek-based name
Category:Powers of 2
Category:Linear omega level