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Fish number 5

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Fish number 5 (F5) is a number defined by Japanese googologist Fish in 2003.[1] It is one of Fish numbers.

Fish function 5 uses m(n) map.

Definition and growth rate of Fish function 5, \(F_5(x)\), is \begin{eqnarray*} F_5(x) & := & ((..((m(x)m(x-1))m(x-2))...m(2))m(1))(x) \\ F_5(x) & \approx & f_{\varepsilon_0}(x) \end{eqnarray*}

Then Fish number 5 is defined as: \[F_5 := F_5^{63}(3)\]

Therefore, Fish number 5 is greater than Fish number 3 and is approximately \(f_{\varepsilon_0+1}(63)\).

Approximations in other notations Edit

Fish number 5 is comparable to Grantethrathoth \(\approx f_{\epsilon_0+1}(100)\).

Notation Approximation
BEAF \(\{63,63,2 (X \uparrow\uparrow X) 2\}\)[2]
Cascading-E Notation E63#^^#63#63
Bird's array notation \(\{63,63,2 [1 \backslash 2] 2\}\)
Hyperfactorial array notation \(63![2,1,1,1,2]\)
Fast-growing hierarchy \(f_{\epsilon_0+1}(63)\)
Hardy hierarchy \(H_{\epsilon_0 \omega}(63)\)
Slow-growing hierarchy \(g_{\vartheta(\varepsilon_{\Omega+1}+1)}(63)\)

Sources Edit

  1. Fish, Googology in Japan - exploring large numbers (2013)
  2. Using particular notation \(\{a,b (X \uparrow\uparrow X) 2\}\) for \(X \uparrow\uparrow b \&\ a\)

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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