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Fish number 1

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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 \(S_z(x,y) = A(x,y,z)\).

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

There is a program[2] for calculating Fish number 1.

Original definition Edit

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 \(g_{\varphi(1,0,0,0)}(63)\)

Sources Edit

  1. Fish, Googology in Japan - exploring large numbers (2013)
  2. Ruby program for calculating Fish number 1

See also Edit

Googology in Asia

Fish numbers: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7
Mapping functions: S map · SS map · S(n) map · M(n) map · M(m,n) map
By Aeton: Okojo numbers · N-growing hierarchy
By BashicuHyudora: Pair sequence number · Bashicu matrix system
Indian counting system: Lakh · Crore
Chinese and Japanese counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
Buddhist text: Tallakshana · Dvajagravati · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra
Other: Taro's multivariable Ackermann function · Sushi Kokuuhen

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