11,329
pages

The penteto is equal to s(5,5,4) in strong array notation.[1] The term was coined by wiki user Hyp cos. It can be represented in up-arrow notation as $$5\uparrow\uparrow\uparrow\uparrow5$$ (5 hexated to 5).

This number is also equal to s(5,2,5) and s(5,4,1,2).

## Etymology

The name of this number is based on the word "penta" and "tetra" (Greek for 5 and 4), meaning a base-5 number, generated using hyper-4 operation based on exponentiation.

## Computation

Penteto can be computed in the following process:

• $$f(1)=5$$
• $$f(2)=5^{5^{5^{5^5}}}$$ with $$f(1)$$ 5's, a.k.a. pentbo
• $$f(3)=5^{5^{\vdots^{5^5}}}$$ with $$f(2)$$ 5's
• $$f(4)=5^{5^{\vdots^{5^5}}}$$ with $$f(3)$$ 5's
• etc.
• Penteto equals $$f^{4}(5)$$.

## Approximations

Notation Approximation
Arrow notation $$5\uparrow\uparrow\uparrow\uparrow5$$ (exact)
BEAF $$\{5,5,4\}$$ (exact)
Hyper-E notation $$E(5)1\#1\#1\#5$$ (exact)
Chained arrow notation $$5\rightarrow5\rightarrow4$$ (exact)
Hyperfactorial array notation $$6!3$$
Fast-growing hierarchy $$f_5(4)$$
Hardy hierarchy $$H_{\omega^5}(4)$$
Slow-growing hierarchy $$g_{\varphi(3,0)}(5)$$