Fish number 2

Fish number 2 (F2) is a number defined by Japanese googologist Fish in 2002.Fish, Googology in Japan - exploring large numbers (2013) Its base is the same as in Fish number 1:
*\(S_1(0,y) = y+1\)
*\(S_z(0,y) = S_{z-1}(y,y)\) for \(z ≥ 1\)
*\(S_z(x,0) = S_z(x - 1,1)\)
*\(S_z(x,y) = S_z(x - 1,S_z(x,y - 1))\)
The function \(S_x(x,x)\) grows about as fast as \(f_{\omega^2}(x)\).
After that, we get a SS map. We have a function \(SS(3,x+1,S)\) which expands to \(S_{x+1}(3,3)\). Then we have an S map over that, which iterates the function \(x+1\). If we have an SS map over that, we get a function with growth rate \(f_{\omega^22}(x)\). Fish number 2 is defined as \(SS^{63}(3,x+1,S)\)
Original definition of Fish number 2 is actually slightly different from above and therefore the number is slightly different, although the order of the number approximated in FGH is similar. The definition is so complex and few people understood it when it was published. After discussions on BBS, the next version of Fish number, Fish number 3 was published, where the definition became simple and the number became larger. Therefore, after publication of \(F_3\), \(F_1\) and \(F_2\) have only historical meaning.
== Approximations ==
Fish showed in his book (pp. 102-103) that S map and SS map are comparable to Taro's 4-variable Ackermann function; when we apply \(a\) times of SS application and \(b\) times of the initial S application to a function f(n)=n+1 (pair of a certain number and this function), the function obtained exactly matchs \(A(a,b,0,n)\). As the definition of Fish number 2 applies SS map 63 times, and get a function is comparable to \(A(63,0,0,n)\). Therefore Fish number 2 is comparable to \(A(1,0,0,0,63) \approx A(63, 0, 0, 63)\).
Notation 	Approximation
BEAF 	\(\{10,10,10,10,63\}\)
Hyper-E notation 	\(E100\#\#\#\#63\)
Chained arrow notation 	\(100 \rightarrow_{63} 100\)
Taro's multivariable Ackermann function	\(A(1,0,0,0,63)\)
s(n) map 	\(s(4)[x+1](63)\)
m(n) map 	\(m(3)^3m(2)m(1)(63)\)
Fast-growing hierarchy 	\(f_{\omega^3}(63)\)
Hardy hierarchy 	\(H_{\omega^{\omega^3}}(63)\)
Slow-growing hierarchy 	\(g_{\varphi(\omega,0,0,0)}(63)\)
== Computer programs ==
There are computer programs for calculating Fish number 2 and its old version:
* aycabta, Ruby program for calculating Fish number 2
* Okkuu, C++ program for calculating Fish number 2
* Okkuu, C++ program for calculating an old version of Fish number 2
* Mitsuki1729, scrach program for calculation Fish number 2
== Sources ==
== See also ==
ja:ふぃっしゅ数バージョン2
Category:Numbers
Category:Googology in Asia
Category:Non-powers
Category:Quadratic omega level
Category:Fish number
Category:Numbers by Fish
Category:Eponymous numbers