Fish number 5

Fish number 5 (F5) is a number defined by Japanese googologist Fish in 2003.Fish, Googology in Japan - exploring large numbers (2013) 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, there are computer programs by Japanese googologists 猫山にゃん太 and Okkuu for calculating Fish number 1, and hence people can understand the behaviour, even if they do not have sufficient knowledge of functions. 
* 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.
* 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)
* Okkuu, ふぃっしゅ数バージョン5を計算するC++プログラム, paiza. (C++ code)
In particular, Fish number version 5 is computable.
== Approximations in other notations ==
Fish number 5 is comparable to grantethrathoth \(\approx f_{\varepsilon_0+1}(100)\).
Notation 	Approximation
BEAF 	\(\{63,63,2 (X \uparrow\uparrow X) 2\}\)Using particular notation \(\{a,b (X \uparrow\uparrow X) 2\}\) for \(X \uparrow\uparrow b \&\ a\)
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 	\(f_{\varepsilon_0+1}(63)\)
Hardy hierarchy 	\(H_{\varepsilon_0 \omega}(63)\)
Slow-growing hierarchy (using Buchholz's function)	\(g_{\psi_0(\Omega_2 + \psi_1(\Omega_2))}(63)\)
== Sources ==
== See also ==
ja:ふぃっしゅ数バージョン5
fr:Cinquième nombre de Fish
Category:Numbers
Category:Googology in Asia
Category:Epsilon level
Category:Fish number
Category:Numbers by Fish
Category:Eponymous numbers