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

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

Community content is available under CC-BY-SA unless otherwise noted.