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Fish number 5 (F5) is a number defined by Japanese googologist Fish in 2003.[1] 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 $$f_{\varepsilon_0+1}(63)$$.

## Approximations in other notations Edit

Fish number 5 is comparable to Grantethrathoth $$\approx f_{\epsilon_0+1}(100)$$.

Notation Approximation
BEAF $$\{63,63,2 (X \uparrow\uparrow X) 2\}$$[2]
Bird's array notation $$\{63,63,2 [1 \backslash 2] 2\}$$
Hyperfactorial array notation $$63![2,1,1,1,2]$$
Fast-growing hierarchy $$f_{\epsilon_0+1}(63)$$
Hardy hierarchy $$H_{\epsilon_0 \omega}(63)$$
Slow-growing hierarchy $$g_{\vartheta(\varepsilon_{\Omega+1}+1)}(63)$$

## Sources Edit

1. Fish, Googology in Japan - exploring large numbers (2013)
2. Using particular notation $$\{a,b (X \uparrow\uparrow X) 2\}$$ for $$X \uparrow\uparrow b \&\ a$$