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

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 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 \(g_{\vartheta(\varepsilon_{\Omega+1}+1)}(63)\)

Sources

  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

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