## FANDOM

10,220 Pages

Fish number 1 (F1), the smallest of the seven Fish numbers, is a number defined by Japanese googologist Fish in 2002.[1] It uses modified Ackermann function, defined as the following:

• $$S_1(0,y) = y+1$$
• $$S_1(x,0) = S_1(x - 1,1)$$
• $$S_1(x,y) = S_1(x - 1,S_1(x,y - 1))$$

After that, we define the $$S_2$$ function that uses $$S_1$$ as the base function and in general, we have an $$S_n$$ function that uses $$S_{n-1}$$ as the base:

• $$S_1(0,y) = y+1$$
• $$S_z(0,y) = S_{z-1}(y,y)$$ for $$z > 1$$
• $$S_z(x,0) = S_z(x - 1,1)$$
• $$S_z(x,y) = S_z(x - 1,S_z(x,y - 1))$$

The Fish function $$F_1(x) = S_x(x,x)$$ grows about as fast as $$f_{\omega^2}(x)$$. The Fish function is equivalent to Taro's 3-variable Ackermann function, namely $$S_z(x,y) = A(z,x,y)$$.

The Fish number 1 is defined with "SS map", which iterates S map, and calculated with $$SS^{63}(3,x+1,S)$$.

Aycabta has created a Ruby program for calculating Fish number 1.[2]

## Original definition Edit

The original definition of Fish number 1 is actually more complicated than above, and the number is also slightly different. The order of the number in FGH is same. Here is the original definition.

[1] Define S map, a map from "a pair of number and function" to "a pair of number and function", as follows. \begin{eqnarray*} S:[m,f(x)]→[g(m),g(x)] \end{eqnarray*}

Here, $$g(x)$$ is given as follows. \begin{eqnarray*} B(0,n) & = & f(n) \\ B(m+1,0) & = & B(m, 1) \\ B(m+1,n+1) & = & B(m, B(m+1, n)) \\ g(x) & = & B(x,x) \end{eqnarray*}

[2] Define SS map, a map from "a set of number, function and S map" to "a set of number, function and S map" as follows.

\begin{eqnarray*} SS:[m,f(x),S]→[n,g(x),S2] \end{eqnarray*}

Here, $$S2$$, $$n$$, and $$g(x)$$ are given as follows. \begin{eqnarray*} S2 & = & S^{f(m)} \\ S2 & : & [m,f(x)]→[n,g(x)] \end{eqnarray*}

[3] Apply SS map 63 times to [3,x+1,S] and we get Fish number $$F_1$$ and Fish function $$F_1(x)$$.

## Approximations in other notations Edit

Fish number 1 is comparable to (slightly larger than) $$A(1,0,1,63)$$ in Taro's 4-variable Ackermann function. Therefore, it is in the order of $$f_{\omega^2+1}(63)$$ in FGH. Thrangol is in the order of $$f_{\omega^2+1}(100)$$, and therefore slightly larger than Fish number 1. Using the approximations in Thrangol, Fish number 1 can be approximated as follows.

Notation Approximation
Chained arrow notation $$3 \rightarrow_2 63 \rightarrow_2 2$$
Notation Array Notation $$(63,2\{4,2\}2)$$
BEAF $$\{4,64,1,1,2\}$$
Extended Hyper-E Notation $$E63\#\#\#63\#\#2$$
Hyperfactorial array notation $$63![2,1,2]$$
Taro's multivariable Ackermann function $$A(1,0,1,63)$$
s(n) map $$s(1)s(3)[x+1](63)$$
m(n) map $$m(2)[m(3)^2m(2)]m(1)(63)$$
Fast-growing hierarchy $$f_{\omega^2+1}(63)$$
Hardy hierarchy $$H_{\omega^{\omega^2+1}}(63)$$
Slow-growing hierarchy $$g_{\varphi(1,0,0,0)}(63)$$