| rdfs:comment | - Fish number 1 (F1), the smallest of the seven Fish numbers, is a number defined by Japanese googologist Fish in 2002. It uses modified Ackermann function, defined as the following:
* \(S_1(0,y) = y+1\)
* \(S_1(x,0) = S_1(x - 1,1)\)
* \(S_1(x,y) = S_1(x - 1,S_1(x,y - 1))\) After that, we define the \(S_2\) function that uses \(S_1\) as the base function and in general, we have an \(S_n\) function that uses \(S_{n-1}\) as the base:
* \(S_1(0,y) = y+1\)
* \(S_z(0,y) = S_{z-1}(y,y)\) for \(z > 1\)
* \(S_z(x,0) = S_z(x - 1,1)\)
* \(S_z(x,y) = S_z(x - 1,S_z(x,y - 1))\)
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| abstract | - Fish number 1 (F1), the smallest of the seven Fish numbers, is a number defined by Japanese googologist Fish in 2002. It uses modified Ackermann function, defined as the following:
* \(S_1(0,y) = y+1\)
* \(S_1(x,0) = S_1(x - 1,1)\)
* \(S_1(x,y) = S_1(x - 1,S_1(x,y - 1))\) After that, we define the \(S_2\) function that uses \(S_1\) as the base function and in general, we have an \(S_n\) function that uses \(S_{n-1}\) as the base:
* \(S_1(0,y) = y+1\)
* \(S_z(0,y) = S_{z-1}(y,y)\) 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 Fish function \(F_1(x) = S_x(x,x)\) grows about as fast as \(f_{\omega^2}(x)\). The Fish function is equivalent to Taro's 3-variable Ackermann function, namely \(S_z(x,y) = A(x,y,z)\). The Fish number 1 is defined with "SS map", which iterates S map, and calculated with \(SS^{63}(3,x+1,S)\). There is a program for calculating Fish number 1.
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