Fish number 3

Fish number 3 (F3) is a number defined by Japanese googologist Fish in 2002.Fish, Googology in Japan - exploring large numbers (2013)Fish "巨大数の世界 (The world of googology)" 数学セミナー (Mathematics seminar) July, 2019. pp. 28--31. 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 , 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,:ja:ふぃっしゅ数バージョン3#計算 (Japanese) 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.Tetramur's post on talk page 17 July 2022
== Approximations in other notations ==
Fish number 3 is smaller than and comparable to godthroogahlah \(\approx f_{(\omega^{\omega+1}) 99}(100)\).
Notation 	Approximation
BEAF 	\(\{63,63 (1) 1,63\}\)
Bird's array notation 	\(\{63,63 [2] 1,63\}\)
Cascading-E Notation 	\(E63\#^\#*\#\#63\)
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 (using Buchholz hierarchy)	\(g_{\psi_0(\Omega^{\Omega^{\Omega + 1} \omega})}(63)\)
== Computer programs ==
There are computer programs for calculating Fish number 3 and its old version:
* aycabta, Ruby program for calculating Fish number 3
* Okkuu, C++ program for calculating Fish number 3
* Okkuu, C++ program for calculating an old version of Fish number 3
== Sources ==
== See also ==
ja:ふぃっしゅ数バージョン3
Category:Numbers
Category:Googology in Asia
Category:Non-powers
Category:Exponentiated linear omega level
Category:Fish number
Category:Numbers by Fish
Category:Eponymous numbers
Category:Numbers with peer-reviewed source