Script error: No such module "Draft topics". Script error: No such module "AfC topic".
Bowers's 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)
The Bowers notation for a linear array includes the following rules:
{
a
}
=
a
{\displaystyle \{a\}=a}
{
a
,
b
}
=
a
b
{\displaystyle \{a,b\}=a^{b}}
{
a
,
b
,
c
,
…
,
n
,
1
}
=
{
a
,
b
,
c
,
…
,
n
}
{\displaystyle \{a,b,c,\ldots ,n,1\}=\{a,b,c,\ldots ,n\}}
{
a
,
1
,
b
,
c
,
…
,
n
}
=
a
{\displaystyle \{a,1,b,c,\ldots ,n\}=a}
{
a
,
b
,
1
,
…
,
1
,
c
,
d
,
…
,
n
}
=
{
a
,
a
,
a
,
…
,
{
a
,
b
−
1
,
1
,
…
,
1
,
c
,
d
,
…
,
n
}
,
c
−
1
,
d
,
…
,
n
}
{\displaystyle \{a,b,1,\ldots ,1,c,d,\ldots ,n\}=\{a,a,a,\ldots ,\{a,b-1,1,\ldots ,1,c,d,\ldots ,n\},c-1,d,\ldots ,n\}}
If rules 1-4 do not apply,
{
a
,
b
,
c
,
d
,
…
,
n
}
=
{
a
,
{
a
,
b
−
1
,
c
,
d
,
…
,
n
}
,
c
−
1
,
d
,
…
,
n
}
{\displaystyle \{a,b,c,d,\ldots ,n\}=\{a,\{a,b-1,c,d,\ldots ,n\},c-1,d,\ldots ,n\}}
Examples [ edit ] The array includes 2 elements
{
10
,
100
}
=
10
100
=
10
↑
100
{\displaystyle \{10,100\}=10^{100}=10\uparrow 100}
The array includes 3 elements
{
10
,
100
,
1
}
=
{
10
,
100
}
{\displaystyle \{10,100,1\}=\{10,100\}}
{
10
,
100
,
2
}
=
{
10
,
{
10
,
99
,
2
}
}
=
{
10
,
{
10
,
{
10
,
98
,
2
}
}
}
=
10
10
10
⋯
10
10
⏟
100
t
e
n
s
=
10
↑↑
100
{\displaystyle \{10,100,2\}=\{10,\{10,99,2\}\}=\{10,\{10,\{10,98,2\}\}\}=\underbrace {10^{10^{10^{\cdots ^{10^{10}}}}}} _{100tens}=10\uparrow \uparrow 100}
{
10
,
100
,
3
}
=
{
10
,
{
10
,
99
,
3
}
,
2
}
=
{
10
,
{
10
,
{
10
,
98
,
3
}
,
2
}
,
2
}
=
10
↑↑↑
100
{\displaystyle \{10,100,3\}=\{10,\{10,99,3\},2\}=\{10,\{10,\{10,98,3\},2\},2\}=10\uparrow \uparrow \uparrow 100}
In general, for a three-element array,
{
a
,
b
,
m
}
=
a
↑
m
b
{\displaystyle \{a,b,m\}=a\uparrow ^{m}b}
Knuth's up-arrow notation .
The array includes 4 elements
{
10
,
100
,
1
,
1
}
=
{
10
,
100
}
{\displaystyle \{10,100,1,1\}=\{10,100\}}
{
10
,
100
,
1
,
2
}
=
{
10
,
10
,
{
10
,
99
,
1
,
2
}
}
=
{
10
,
10
,
{
10
,
10
,
{
10
,
98
,
1
,
2
}
}
}
=
10
↑↑
⋯
↑↑
10
⏟
10
↑↑
⋯
↑↑
10
⏟
⋮
⏟
10
↑↑
⋯
↑↑
10
⏟
10 arrows
}
100
≈
10
→
10
→
100
→
2
{\displaystyle \{10,100,1,2\}=\{10,10,\{10,99,1,2\}\}=\{10,10,\{10,10,\{10,98,1,2\}\}\}=\left.{\begin{matrix}&&\underbrace {10\uparrow \uparrow \cdots \uparrow \uparrow 10} \\&&\underbrace {10\uparrow \uparrow \cdots \uparrow \uparrow 10} \\&&\underbrace {\qquad \ \;\;\vdots \qquad \;\;} \\&&\underbrace {10\uparrow \uparrow \cdots \uparrow \uparrow 10} \\&&{\text{10 arrows}}\end{matrix}}\right\}{\text{100 }}\approx 10\rightarrow 10\rightarrow 100\rightarrow 2}
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
}
a
p
p
r
o
x
10
→
10
→
100
→
3
{\displaystyle \{10,100,2,2\}=\{10,\{10,99,2,2\},1,2\}=\{10,\{10,\{10,98,2,2\},1,2\},1,2\}\ approx10\rightarrow 10\rightarrow 100\rightarrow 3}
{
10
,
100
,
m
,
2
}
≈
10
→
10
→
100
→
(
m
+
1
)
{\displaystyle \{10,100,m,2\}\approx 10\rightarrow 10\rightarrow 100\rightarrow (m+1)}
In general, for a four-element array, the following is true
{
a
,
b
,
c
,
d
}
>
a
→
a
→
⋯
a
→
a
⏟
d
−
1
arrow
→
(
b
−
1
)
→
(
c
+
1
)
{\displaystyle \{a,b,c,d\}>\underbrace {a\rightarrow a\rightarrow \cdots a\rightarrow a} _{d-1{\text{arrow}}}\rightarrow (b-1)\rightarrow (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
ω
{\displaystyle \omega }
ω
2
{\displaystyle \omega ^{2}}
ω
ω
{\displaystyle \omega ^{\omega }}
fast-growing hierarchy .
Examples in Expression
Related
This article "Bowers's operators" is from Wikipedia . The list of its authors can be seen in its historical and/or the page Edithistory:Bowers's operators . Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.
