Fish number 5 (F5) is a number defined by Japanese googologist Fish in 2003. 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, a Japanese Googologist 猫山にゃん太 formulated an alternative computable notation equivalent to this system, and hence people can understand the behaviour, even if they do not have sufficient knowledge of functions. In particular, Fish number version 5 is computable.

## Approximations in other notations

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

Notation Approximation
BEAF $$\{63,63,2 (X \uparrow\uparrow X) 2\}$$
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

1. Fish, Googology in Japan - exploring large numbers (2013)
2. 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.
3. 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)
4. Using particular notation $$\{a,b (X \uparrow\uparrow X) 2\}$$ for $$X \uparrow\uparrow b \&\ a$$

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