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Googology Wiki

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 \(f_{\varepsilon_0+1}(63)\).


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 5 requires a deep understanding of the notions of functions. On the other hand, a Japanese Googologist 猫山にゃん太 formulated an alternative computable notation equivalent to this system,[2][3] and hence people can understand the behaviour, even if they do not have sufficient knowledge of functions. In particular, Fish number version 5 is computable.

Approximations in other notations

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

Notation Approximation
BEAF \(\{63,63,2 (X \uparrow\uparrow X) 2\}\)[4]
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 \(g_{\vartheta(\varepsilon_{\Omega+1}+1)}(63)\)


  1. Fish, Googology in Japan - exploring large numbers (2013)
  2. 猫山にゃん太, ふぃっしゅ数バージョン5の機械的展開方法, Japanese Googology Wiki user blog.
  3. 猫山にゃん太, Fish Number 5 - ふぃっしゅ数バージョン 5, GitHub page. (javascript code)
  4. Using particular notation \(\{a,b (X \uparrow\uparrow X) 2\}\) for \(X \uparrow\uparrow b \&\ a\)

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