Knuth's up-arrow notation

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, Knuth's up-arrow notation is a method of notation for <a href="/facts/Large_number/FgYabI0S">very large</a> <a href="/facts/Integer/PUwwrawx">integers</a>, introduced by <a href="/facts/Donald_Knuth/GozNzDM6">Donald Knuth</a> in 1976.
In his 1947 paper, <a href="/facts/R._L._Goodstein/7sSAxMPt">R. L. Goodstein</a> introduced the specific sequence of operations that are now called <a href="/facts/Hyperoperation/XNn1hneN">hyperoperations</a>. Goodstein also suggested the Greek names <a href="/facts/Tetration/5rKeVVYv">tetration</a>, <a href="/facts/Pentation/YKhnIWPG">pentation</a>, etc., for the extended operations beyond <a href="/facts/Exponentiation/pac6QY2g">exponentiation</a>. The sequence starts with a <a href="/facts/Unary_operation/kb6CkgsX">unary operation</a> (the <a href="/facts/Successor_function/R1hhkSEG">successor function</a> with n = 0), and continues with the <a href="/facts/Binary_operation/CYtow0bz">binary operations</a> of <a href="/facts/Addition/apFWJBFB">addition</a> (n = 1), <a href="/facts/Multiplication/VtcUjpNv">multiplication</a> (n = 2), <a href="/facts/Exponentiation/pac6QY2g">exponentiation</a> (n = 3), <a href="/facts/Tetration/5rKeVVYv">tetration</a> (n = 4), <a href="/facts/Pentation/YKhnIWPG">pentation</a> (n = 5), etc.
<a href="/facts/Hyperoperation/XNn1hneN">Various notations</a> have been used to represent hyperoperations. One such notation is 
 
 
 
 
 H
 
 n
 
 
 (
 a
 ,
 b
 )
 
 
 {\displaystyle H_{n}(a,b)}
 
.
Knuth's up-arrow notation 
 
 
 
 ↑
 
 
 {\displaystyle \uparrow }
 
 is another. 
For example:

<ul><li>the single arrow 
 
 
 
 ↑
 
 
 {\displaystyle \uparrow }
 
 represents <a href="/facts/Exponentiation/pac6QY2g">exponentiation</a> (iterated multiplication) 
 
 
 
 2
 ↑
 4
 =
 
 H
 
 3
 
 
 (
 2
 ,
 4
 )
 =
 2
 ×
 (
 2
 ×
 (
 2
 ×
 2
 )
 )
 =
 
 2
 
 4
 
 
 =
 16
 
 
 {\displaystyle 2\uparrow 4=H_{3}(2,4)=2\times (2\times (2\times 2))=2^{4}=16}
 
</li>
<li>the double arrow 
 
 
 
 ↑↑
 
 
 {\displaystyle \uparrow \uparrow }
 
 represents <a href="/facts/Tetration/5rKeVVYv">tetration</a> (iterated exponentiation) 
 
 
 
 2
 ↑↑
 4
 =
 
 H
 
 4
 
 
 (
 2
 ,
 4
 )
 =
 2
 ↑
 (
 2
 ↑
 (
 2
 ↑
 2
 )
 )
 =
 
 2
 
 
 2
 
 
 2
 
 2
 
 
 
 
 
 
 =
 
 2
 
 16
 
 
 =
 65
 ,
 536
 
 
 {\displaystyle 2\uparrow \uparrow 4=H_{4}(2,4)=2\uparrow (2\uparrow (2\uparrow 2))=2^{2^{2^{2}}}=2^{16}=65,536}
 
</li>
<li>the triple arrow 
 
 
 
 ↑↑↑
 
 
 {\displaystyle \uparrow \uparrow \uparrow }
 
 represents <a href="/facts/Pentation/YKhnIWPG">pentation</a> (iterated tetration) 
 
 
 
 
 
 
 
 2
 ↑↑↑
 4
 =
 
 H
 
 5
 
 
 (
 2
 ,
 4
 )
 =
 2
 ↑↑
 (
 2
 ↑↑
 (
 2
 ↑↑
 2
 )
 )
 
 
 
 
 
 
 =
 2
 ↑↑
 (
 2
 ↑↑
 (
 2
 ↑
 2
 )
 )
 
 
 
 
 
 
 =
 2
 ↑↑
 (
 2
 ↑↑
 4
 )
 
 
 
 
 
 
 =
 
 
 
 2
 ↑
 (
 2
 ↑
 (
 2
 ↑
 …
 )
 )
 
 ⏟
 
 
 
 =
 
 
 
 
 
 
 2
 
 
 2
 
 
 ⋯
 
 2
 
 
 
 
 
 
 
 ⏟
 
 
 
 
 
 
 
 
 
 
 
 
 2
 ↑↑
 4
 
 
  copies of 
 
 
 2
 
 
 
 
 
 
 
 65,536 2s
 
 
 
 
 
 
 
 
 {\displaystyle {\begin{aligned}2\uparrow \uparrow \uparrow 4=H_{5}(2,4)=2\uparrow \uparrow (2\uparrow \uparrow (2\uparrow \uparrow 2))\\&=2\uparrow \uparrow (2\uparrow \uparrow (2\uparrow 2))\\&=2\uparrow \uparrow (2\uparrow \uparrow 4)\\&=\underbrace {2\uparrow (2\uparrow (2\uparrow \dots ))} \;=\;\underbrace {\;2^{2^{\cdots ^{2}}}} \\&\;\;\;\;\;2\uparrow \uparrow 4{\mbox{ copies of }}2\;\;\;\;\;{\mbox{65,536 2s}}\\\end{aligned}}}
 
</li></ul>
The general definition of the up-arrow notation is as follows (for 
 
 
 
 a
 ≥
 0
 ,
 n
 ≥
 1
 ,
 b
 ≥
 0
 
 
 {\displaystyle a\geq 0,n\geq 1,b\geq 0}
 
):

a
        
          ↑
          
            n
          
        
        b
        =
        
          H
          
            n
            +
            2
          
        
        (
        a
        ,
        b
        )
        =
        a
        [
        n
        +
        2
        ]
        b
        .
      
    
    {\displaystyle a\uparrow ^{n}b=H_{n+2}(a,b)=a[n+2]b.}

Here, 
 
 
 
 
 ↑
 
 n
 
 
 
 
 {\displaystyle \uparrow ^{n}}
 
 stands for n arrows, so for example

2
        ↑↑↑↑
        3
        =
        2
        
          ↑
          
            4
          
        
        3.
      
    
    {\displaystyle 2\uparrow \uparrow \uparrow \uparrow 3=2\uparrow ^{4}3.}

The square brackets are another notation for hyperoperations.

Knuth's up-arrow notation open-in-new

Knuth's up-arrow notation