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 \(f_{\zeta_0+1}(63)\).

Approximations

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 \(H_{\zeta_0 \omega}(63)\)

Sources

  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

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: Primitive sequence number · Pair sequence number · Bashicu matrix system
By Kanrokoti: KumaKuma ψ function
By 巨大数大好きbot: Flan numbers
By Jason: Irrational arrow notation · δOCF · δφ · ε function
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 Nayuta Ito: N primitive
By p進大好きbot: Large Number Garden Number
By Yukito: Hyper primitive sequence system · Y sequence · YY sequence · Y function
Indian counting system: Lakh · Crore · Tallakshana · Uppala · Dvajagravati · Paduma · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra · Nirabhilapya nirabhilapya parivarta
Chinese, Japanese and Korean counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
Other: Taro's multivariable Ackermann function · TR function · Arai's \(\psi\) · Sushi Kokuu Hen

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