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For the number {4,4,4,4}, see supertet.

Conway's Tetratet, or Conway's four-four-four-four, is a number mentioned by John H. Conway in the Book of Numbers.[1] It is the largest explicitly defined finite number in the entire book. It is also the 4th Conway number.[2]

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 Edit

Notation Approximation
BEAF $$\{4,5,3,2\}$$
Fast-growing hierarchy $$f_{\omega+2}(f_{\omega+2}(f_{\omega+2}(255))$$
Hardy hierarchy $$H_{\omega^{\omega+2}3}(255)$$
Slow-growing hierarchy $$g_{\varphi(1,1,0)}(255)$$
Hyperfactorial array notation $$4![1,2] = 4![4]$$
Boris's sezration notation $$[4,4]$$
Notation array $$(4\{3,4\}4)$$