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Boogafive is equal to $$5\uparrow\uparrow\uparrow5$$, or 5 pentated to 5.[1]

Googology Wiki user Hyp cos calls this number pentro, and it's equal to s(5,5,3) = s(5,2,4) = s(5,3,1,2) in strong array notation.[2]

## Etymology

The name "boogafive" is formed by adding the booga- prefix to 5.

The name "pentro" is based on the word "penta" and "tri" (Greek for 5 and 3), meaning a base-5 number, generated using hyper-3 operation based on exponentiation.

## Computation

Boogafive can be computed in the following process:

• $$f(1)=5$$
• $$f(2)=5^5=3,125$$
• $$f(3)=5^{5^5}\approx1.911\times10^{2,184}$$
• $$f(4)=5^{5^{5^5}}$$
• etc.
• Boogafive is equal to $$f(f(f(f(5))))$$.

## Approximations

Notation Approximation
Arrow notation $$5\uparrow\uparrow\uparrow5$$ (exact)
BEAF $$\{5,5,3\}$$ (exact)
Hyper-E notation $$E(5)1\#1\#5$$ (exact)
Chained arrow notation $$5\rightarrow5\rightarrow3$$ (exact)
Hyperfactorial array notation $$6!2$$
Fast-growing hierarchy $$f_4(4)$$
Hardy hierarchy $$H_{\omega^4}(4)$$
Slow-growing hierarchy $$g_{\zeta_0}(5)$$