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Fish number 6 (F6) is a number defined by Japanese googologist Fish in 2007.[1] It is largest of the computable Fish numbers.

Fish function 6 uses m(m,n) map.

Definition and growth rate of Fish function 6, \(F_6(x)\), is

\begin{eqnarray*} F_6(x) &:=& m(x,2)m(x,1)(x) \\ F_6(x) &\approx& f_{\zeta_0}(x) \end{eqnarray*}

Then Fish number 6 is defined as: \[F_6 := F_6^{63}(3)\]

Therefore, Fish number 6 is greater than Fish number 5 and is approximately \(f_{\zeta_0+1}(63)\).


Notation Approximation
BEAF \(\{63,63,2 (X \uparrow\uparrow X \uparrow 2) 2\}\)[2]
Pair sequence (0,0)(1,1)(2,1)(1,0)[8]
Fast-growing hierarchy \(f_{\zeta_0+1}(63)\)
Hardy hierarchy \(H_{\zeta_0 \omega}(63)\)


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

See also

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's \(\psi\)
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4
By ふぃっしゅ (Fish): Fish numbers (Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7 · S map · SS map · s(n) map · m(n) map · m(m,n) map) · Bashicu matrix system 1/2/3/4 computation programmes · TR function (I0 function)
By じぇいそん (Jason): Irrational arrow notation · δOCF · δφ · ε function
By 甘露東風 (Kanrokoti): KumaKuma ψ function
By 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
By koteitan: Bashicu matrix system 2.3
By mrna: 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ
By Naruyoko Naruyo: Y sequence computation programme · ω-Y sequence computation programme
By Nayuta Ito: N primitive · Flan numbers · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4
By p進大好きbot: Large Number Garden Number
By たろう (Taro): Taro's multivariable Ackermann function
By ゆきと (Yukito): Hyper primitive sequence system · Y sequence · YY sequence · Y function · ω-Y sequence
See also: Template:Googology in Asia