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• Comment: Not sourced for notability (and nor is Conway for that matter). ChrysGalley (talk) 14:31, 26 April 2026 (UTC)

Array Notation is a means of expressing extremely large numbers, created by Jonathan Bowers^{[1] [2]}, vastly exceeding Conway chained arrow notation, a notation designed to produce very large numbers ^[3] (see Bird's proof).^{[4] [5] [6]}

Let ${\displaystyle a,b,c,\ldots ,a_{1},a_{2},a_{3},\ldots }$ denote positive integers and ${\displaystyle \#}$ denote an arbitrary sequence of positive integers. An entry is a positive integer in an array. An example of a valid array is ${\displaystyle \{4,5,1,3,2\}}$. The array is evaluated according to the following rules:

1. ${\displaystyle \{a\}=a}$
2. ${\displaystyle \{a,b\}=a^{b}}$ (originally ${\displaystyle a+b}$)^[5]
3. ${\displaystyle \{\#,1\}=\{\#\}}$
4. ${\displaystyle \{a,1,\#\}=a}$
5. ${\displaystyle \{a,b,1,\ldots ,1,c,\#\}=\{a,a,\ldots ,a,\{a,b-1,1,\ldots ,1,c,\#\},c-1,\#\}}$ (if the third entry is ${\displaystyle 1}$, all entries before the last ${\displaystyle 1}$ preceding non-${\displaystyle 1}$ entry ${\displaystyle c}$ become the first entry, the last of the ones becomes the original array with the second entry decreased by ${\displaystyle 1}$, and the said non-${\displaystyle 1}$ entry is decreased by ${\displaystyle 1}$)
6. If none of the rules above apply, then ${\displaystyle \{a,b,c,\#\}=\{a,\{a,b-1,c,\#\},c-1,\#\}}$ ${\displaystyle (c>1)}$^[6]

1. The array above is evaluated like so: ${\displaystyle \{4,5,1,3,2\}=\{4,4,\{4,4,1,3,2\},2,2\}}$ (third entry is ${\displaystyle 1}$, rule 5)
2. ${\displaystyle \{5,5,1,1,2\}=\{5,5,5,\{5,4,1,1,2\},1\}=\{5,5,5,\{5,4,1,1,2\}\}=}$
   ${\displaystyle \{5,5,5,\{5,5,5,\{5,3,1,1,2\}\}\}=\{5,5,5,\{5,5,5,\{5,5,5,\{5,2,1,1,2\}\}\}\}=}$
   ${\displaystyle \{5,5,5,\{5,5,5,\{5,5,5,\{5,5,5,\{5,1,1,1,2\}\}\}\}\}=\{5,5,5,\{5,5,5,\{5,5,5,\{5,5,5,5\}\}\}\}}$
3. ${\displaystyle \{2,3,3\}=\{2,\{2,2,3\},2\}=\{2,\{2,\{2,1,3\},2\},2\}=\{2,\{2,2,2\},2\}=\{2,\{2,\{2,1,2\},1\},2\}=}$
   ${\displaystyle \{2,\{2,2,1\},2\}=\{2,\{2,2\},2\}=\{2,4,2\}=\{2,\{2,3,2\},1\}=\{2,\{2,3,2\}\}=}$
   ${\displaystyle \{2,\{2,\{2,2,2\}\}\}=\{2,\{2,\{2,\{2,1,2\}\}\}\}=\{2,\{2,\{2,2\}\}\}=2^{2^{2^{2}}}=2\uparrow \uparrow 4=65536}$
4. ${\displaystyle \{a,b,c\}=a\uparrow ^{c}b}$^[5]

Bowers also created a set of "extended operators" to visualize his array notation.^[6] ${\displaystyle {\begin{array}{lcl}a\{c\}b=a\uparrow ^{c}b=\{a,b,c\}\\a\underbrace {\{\{\ldots \{\{} _{d}c\underbrace {\}\}\ldots \}\}} _{d}b=a\{c\}^{d}b=\{a,b,c,d\}\end{array}}}$

Array notation with 4 or more entries is not primitive recursive because the Ackermann function ${\displaystyle A(n,n)=(2\{n-2\}(n+3))-3(n\geq 3)}$ grows slower than even ${\displaystyle \{n,n,n\}=n\{n\}n}$

Bird’s Proof is a theorem that states that ${\displaystyle \forall a\geq 3,b\geq 2,c\geq 1,d\geq 2,\{a,b,c,d\}>\underbrace {a\rightarrow a\rightarrow a\rightarrow \ldots \rightarrow a} _{d}\rightarrow (b-1)\rightarrow (c+1)}$ in Conway chained arrow notation. The theorem was proven by Chris Bird and named by Jonathan Bowers.^[4][2] Before proving the main theorem, Bird proved two lemmas:

• ${\displaystyle a\rightarrow \#\geq a\forall }$ Conway chains.
• ${\displaystyle a\rightarrow a\rightarrow \ldots \rightarrow a\rightarrow (y+1)\rightarrow z>(a\rightarrow a\rightarrow \ldots \rightarrow a\rightarrow y\rightarrow z)+1}$ (${\displaystyle n}$ ${\displaystyle a}$’s on each side)

and two corollaries:

• ${\displaystyle y'>y\geq 1\implies a\rightarrow a\rightarrow \ldots \rightarrow a\rightarrow y'\rightarrow z>a\rightarrow a\rightarrow \ldots \rightarrow a\rightarrow y\rightarrow z}$

(${\displaystyle n}$ ${\displaystyle a}$’s on each side), ${\displaystyle \forall a\geq 3,z\geq 1,n\geq 1}$.

• ${\displaystyle y'>y\geq 1\implies a\rightarrow a\rightarrow \ldots \rightarrow a\rightarrow y'>a\rightarrow a\rightarrow \ldots \rightarrow a\rightarrow y}$

(${\displaystyle n}$ ${\displaystyle a}$’s on each side), ${\displaystyle \forall a\geq 3,z\geq 1,n\geq 1}$. Note that the second corollary is just the first corollary, but with ${\displaystyle z=1}$.

The main theorem was proven by mathematical induction.

The proof implies that:

1. ${\displaystyle \{a,a,a,d\}>\underbrace {a\rightarrow a\rightarrow a\rightarrow \dots \rightarrow a} _{d+2}\forall a\geq 3,d>1}$, as ${\displaystyle \{a,a,a,1\}=\{a,a,a\}=a\uparrow ^{a}a=a\rightarrow a\rightarrow a}$
2. $${\displaystyle \left.{\begin{matrix}\{a,b,1,1,2\}>&a\underbrace {\rightarrow a\rightarrow \ldots \rightarrow } a\\&(a\underbrace {\rightarrow a\rightarrow \ldots \rightarrow } a)+2\\&\underbrace {\qquad \;\;\vdots \qquad \;\;} \\&(a\underbrace {\rightarrow a\rightarrow \ldots \rightarrow } a)+2\\&a+2\end{matrix}}\right\}b{\text{ layers}}}$$${\displaystyle \forall a\geq 3}$, ${\displaystyle b\geq 3}$

Graham's number cannot be easily expressed in array notation but can be bounded by the following: ${\displaystyle \{3,65,1,2\}<}$ Graham's number ${\displaystyle <\{4,65,1,2\}\ll \{3,66,1,2\}}$

Let ${\displaystyle f(x)=\{3,3,x\}}$. Then, Graham’s Number is ${\displaystyle f^{64}(4)}$, where the superscript represents iteration. Since ${\displaystyle \{3,3,3\}=f(3)}$, it follows that ${\displaystyle f^{64}(3)=\{3,65,1,2\}<f^{64}(4)=\{3,3,f^{63}(4)\}=}$ Graham’s Number ${\displaystyle <\{4,4,f^{63}(4)\}<\{4,65,1,2\}\ll \{3,66,1,2\}}$

Bowers created a simple function: ${\displaystyle b\&a}$ (${\displaystyle b}$ array of ${\displaystyle a}$) ${\displaystyle =\underbrace {\{a,a,a,\ldots ,a\}} _{b}}$. As one might expect, this grows extraordinarily faster than Conway chained arrow notation. (${\displaystyle a\&a\approx f_{\omega ^{\omega }}(a)}$ in the fast-growing hierarchy^[7])

Bowers extended his array notation to more dimensions and even tetrational spaces. It has a growth rate of ${\displaystyle f_{\varepsilon _{0}}(a)}$ in the fast-growing hierarchy.^[2].

${\displaystyle \#,\#_{1},\ldots }$ denote remainders of an array, ${\displaystyle \&}$ denotes the "array of" operator, ${\displaystyle (n),n>0}$ denotes a "separator" and ${\displaystyle (n_{1})(n_{2})\ldots (n_{m})}$ denote separators such that ${\displaystyle n_{1}\geq n_{2}\geq \ldots \geq n_{m}>0}$. The rules only cover cases up to ${\displaystyle f_{\omega ^{\omega ^{\omega }}}(a)}$, but can be easily extended to ${\displaystyle f_{\varepsilon _{0}}(a)}$^[2]:

1. 2 entries: ${\displaystyle \{a,b\}=a^{b}}$
2. Rows ending with 1: ${\displaystyle \{\#1(m)\#_{1}1(n)\#_{2}1\}=\{\#(m)\#_{1}(n)\#_{2}\}}$
3. Second entry is 1: ${\displaystyle \{a,1\#\}=a}$
4. Third entry is 1, non-1 entry in same row ${\displaystyle \{a,b,1,\ldots ,1,c\#\}=\{a,a,a,\ldots ,a,\{a,b-1,1,\ldots ,1,c,\#\},c-1\#\}}$
5. Rules above do not apply: ${\displaystyle \{a,b,c\#\}=\{a,\{a,b-1,c\#\},c-1\#\}}$
6. 2 entries on main row, next non-1 entry begins a higher structure: ${\displaystyle \{a,b(n)\#(k)c\#\}=\{b^{n}\&a(n)\#(k)c-1\#\}}$
7. 2 entries on main row, next non-1 entry is preceded by 1's in its row: ${\displaystyle \{a,b(n)\#(k)1,\ldots ,1,c\#_{1}\}=\{b^{n}\&a(n)\#(k)a,a,\ldots a,\{a,b-1(n)\#(k)1,1,\dots 1,1,c\#_{1}\},c-1\#_{1}\}}$^[2]
8. ${\displaystyle b^{0}\&a=1\&a=a}$
9. ${\displaystyle b^{1}\&a=b\&a}$
10. ${\displaystyle b^{n+1}\&a=\underbrace {b^{n}\&a(n)\ldots (n)b^{n}\&a} _{b}}$ for ${\displaystyle n\geq 1}$