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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*}

## Computation

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.

## Estimation

A Googology Wiki user Tetramur pointed out that Fish number 3 can be exactly expressed via slightly modified Wainer hierarchy-based FGH as $$f'^{63}_{\omega^{\omega+1}63}(3)$$. This version of FGH has only a single modification of fundamental sequences, mainly $$\omega^\omega[n] = \omega^{n-1}$$ instead of $$\omega^\omega[n] = \omega^n$$ in standard Wainer hierachy.[4]

## Approximations in other notations

Fish number 3 is smaller than and comparable to godthroogahlah $$\approx f_{(\omega^{\omega+1}) 99}(100)$$.

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 $$g_{\vartheta((\Omega^{\Omega}) 63 + 1)}(63)$$

## Computer programs

There are computer programs for calculating Fish number 3 and its old version:

## Sources

Original numbers, functions, notations, and notions

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's psi function
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4 original idea
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 formalisation · TR function (I0 function)
By Gaoji: Weak Buchholz's 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 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 formalisation · ω-Y sequence formalisation
By Nayuta Ito: N primitive · Flan numbers (Flan number 1 · Flan number 2 · Flan number 3 · Flan number 4 version 3 · Flan number 5 version 3) · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4
By Okkuu: Extended Weak Buchholz's function
By p進大好きbot: Ordinal notation associated to Extended Weak Buchholz's function · Ordinal notation associated to Extended Buchholz's function · Naruyoko is the great · 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 original idea · YY sequence · Y function · ω-Y sequence original idea

Methodology

By バシク (BashicuHyudora): Bashicu matrix system as a notation template
By じぇいそん (Jason): Shifting definition
By mrna: Side nesting
By Nayuta Ito and ゆきと (Yukito): Difference sequence system

Implementation of existing works into programs

Proofs, translation maps for analysis schema, and other mathematical contributions

By ふぃっしゅ (Fish): Computing last 10000 digits of mega · Translation map for primitive sequence system and Cantor normal form
By Kihara: Proof of an estimation of TREE sequence · Proof of the incomparability of Busy Beaver function and FGH associated to Kleene's $$\mathcal{O}$$
By koteitan: Translation map for primitive sequence system and Cantor normal form
By Naruyoko Naruyo: Translation map for Extended Weak Buchholz's function and Extended Buchholz's function
By Nayuta Ito: Comparison of Steinhaus-Moser Notation and Ampersand Notation
By p進大好きbot: Proof of the termination of Hyper primitive sequence system · Proof of the termination of Pair sequence number · Proof of the termination of segements of TR function in the base theory under the assumption of the $$\Sigma_1$$-soundness and the pointwise well-definedness of $$\textrm{TR}(T,n)$$ for the case where $$T$$ is the formalisation of the base theory

Entertainments

By 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
By koteitan: Dancing video of a Gijinka of Fukashigi · Dancing video of a Gijinka of 久界 · Storyteller's theotre video reading Large Number Garden Number aloud