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Fish number 3 (F3) is a number defined by Japanese googologist Fish in 2002.[1][2] It is one of the seven Fish numbers.

Define s(n) map as: \begin{eqnarray*} s(1)f & := & g; g(x)=f^x(x) \\ s(n)f & := & g; g(x)=[s(n-1)^x]f(x) (\text{if }n>1) \end{eqnarray*}

where s(n) is a functional, and the growth rate in FGH is

\begin{eqnarray*} s(x)f(x) \approx f_{\omega^\omega}(x) \end{eqnarray*}

ss(n) map is defined as: \begin{eqnarray*} ss(1)f & := & g; g(x)=s(x)f(x) \\ ss(n)f & := & g; g(x)=[ss(n-1)^x]f(x) (\text{if }n>1) \\ \end{eqnarray*}

and the growth rate is \begin{eqnarray*} ss(1)f(x) = s(x)f(x) & \approx & f_{\omega^\omega}(x) \\ ss(n)f(x) & \approx & f_{\omega^{\omega+n-1}}(x) \end{eqnarray*}

Definition and growth rate of Fish function 3, \(F_3(x)\), is \begin{eqnarray*} F_3(x) & := & ss(2)^{63}f; f(x)=x+1 \\ F_3(x) & \approx & f_{\omega^{\omega+1}\times63}(x) \end{eqnarray*}

Finally, \begin{eqnarray*} F_3 := F_3^{63}(3) \approx f_{\omega^{\omega+1}\times63 + 1}(63) \end{eqnarray*}


Similar to the systems for other fish numbers, this system uses translations of functions. Therefore, unlike usual systems simply rewriting terms, the understanding of the precise definition of Fish number 3 requires a deep understanding of the notions of functions. On the other hand, p進大好きbot formulated an alternative computable notation equivalent to this system,[3] and hence people can understand the behaviour, even if they do not have sufficient knowledge of functions. In particular, Fish number 3 is computable.

Approximations in other notations

Fish number 3 is comparable to godekagoldgahlah \(\approx f_{(\omega^{\omega+1}) 9}(100)\).

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


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