Bowers's operators

Bowers' operators (BEAF) is a notation for writing large numbers proposed by the American mathematician Jonathan Bowers in 2002. This notation is a generalization of the preceding 4-argument notation (known as Bowers' operators).

Rules

The Bowers notation for a linear array includes the following rules:

1. $\{a\}=a$ and $\{a,b\}=a^b$
2. $\{a,b,c,\dots ,n,1\}=\{a,b,c,\dots ,n\}$
3. $\{a,1,b,c,\dots ,n\}=a$
4. $\{a,b,1,\dots ,1,c,d,\dots ,n\}=\{a,a,a,\dots ,\{a,b-1,1,\dots ,1,c,d,\dots ,n\},c-1,d,\dots ,n\}$.
5. If rules 1-4 do not apply, $\{a,b,c,d,\dots ,n\}=\{a,\{a,b-1,c,d,\dots ,n\},c-1,d,\dots ,n\}$

Examples

The array includes 2 elements

• $\{10,100\}=10^{100}=10\uparrow 100$ (rule 1 applied)

The array includes 3 elements

• $\{10,100,1\}=\{10,100\}$ (rule 2 applied)
• $\{10,100,2\}=\{10,\{10,99,2\}\}=\{10,\{10,\{10,98,2\}\}\}=\underset{100tens}{\underbrace{10^{10^{10^{{\cdots }^{10^{10}}}}}}}=10\uparrow \uparrow 100$ (rule 5 applied)
• $\{10,100,3\}=\{10,\{10,99,3\},2\}=\{10,\{10,\{10,98,3\},2\},2\}=10\uparrow \uparrow \uparrow 100$ (rule 5 applied)

In general, for a three-element array, $\{a,b,m\}=a{\uparrow }^{m}b$ is true according to Knuth's up-arrow notation.

The array includes 4 elements

• $\{10,100,1,1\}=\{10,100\}$ (rule 2 applied)
• $\{10,100,1,2\}=\{10,10,\{10,99,1,2\}\}=\{10,10,\{10,10,\{10,98,1,2\}\}\}=\begin{array}{ccc}& & \underbrace{10\uparrow \uparrow \cdots \uparrow \uparrow 10}\\ & & \underbrace{10\uparrow \uparrow \cdots \uparrow \uparrow 10}\\ & & \underbrace{⋮}\\ & & \underbrace{10\uparrow \uparrow \cdots \uparrow \uparrow 10}\\ & & \text{10 arrows}\end{array}\}\text{100 }\approx 10\to 10\to 100\to 2$ (rule 4 is applied)

and this is already more than Graham number (the Graham number itself is somewhere between {3,64,1,2} and {3,65,1,2}).

• $\{10,100,2,2\}=\{10,\{10,99,2,2\},1,2\}=\{10,\{10,\{10,98,2,2\},1,2\},1,2\}\approx 10\to 10\to 100\to 3$ (rule 5 applied)
• $\{10,100,m,2\}\approx 10\to 10\to 100\to (m+1)$

In general, for a four-element array, the following is true

$\{a,b,c,d\}>\underset{d-1\text{arrow}}{\underbrace{a\to a\to \cdots a\to a}}\to (b-1)\to (c+1)$

according to Conway chained arrow notation.

Thus, if the Bowers array, which includes 3 elements, has the power of Knuth's up-arrow notation (fEdlimit $\omega$), then the four-element array already has the power of Conway notation (limit ${\omega }^2$), and so on with the addition of each new element. The Bowers notation for a linear array including a finite number of elements has a limit ${\omega }^{\omega }$ in the terminology of fast-growing hierarchy.

Notes

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• Large numbers

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