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Fish number 3 (F3) is a number defined by Japanese googologist Fish in 2002.[1] 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 functional, 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*}

## Approximations in other notations Edit

Fish number 3 is comparable to godekagoldgahlah $$\approx f_{(\omega^{\omega+1}) 9}(100)$$.

Notation Approximation
BEAF $$\{63,63 (1) 2,63\}$$
Cascading-E Notation $$E63\#^{\#+2}63$$
Bird's array notation $$\{63,63 [2] 2,63\}$$
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 $$g_{\vartheta((\Omega^{\Omega}) 63 + 1)}(63)$$

## Sources Edit

1. Fish, Googology in Japan - exploring large numbers (2013)