Knuth's up-arrow notation

<h2 id="introduction">Introduction</h2>
<p>The <a href="/facts/Hyperoperations/XNn1hneN">hyperoperations</a> naturally extend the <a href="/facts/Arithmetic/hvHwkbMV">arithmetic</a> operations of <a href="/facts/Addition/apFWJBFB">addition</a> and <a href="/facts/Multiplication/VtcUjpNv">multiplication</a> as follows.
<a href="/facts/Addition/apFWJBFB">Addition</a> by a <a href="/facts/Natural_number/u65og8JE">natural number</a> is defined as iterated incrementation:
</p>

H
                  
                    1
                  
                
                (
                a
                ,
                b
                )
                =
                a
                +
                b
                =
              
              
                a
                +
                
                  
                    
                      1
                      +
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                      +
                      ⋯
                      +
                      1
                    
                    ⏟
                  
                
              
            
            
              
              
                b
                
                  
                     copies of 
                  
                
                1
              
            
          
        
      
    
    {\displaystyle {\begin{matrix}H_{1}(a,b)=a+b=&a+\underbrace {1+1+\dots +1} \\&b{\mbox{ copies of }}1\end{matrix}}}

<p><a href="/facts/Multiplication/VtcUjpNv">Multiplication</a> by a <a href="/facts/Natural_number/u65og8JE">natural number</a> is defined as iterated <a href="/facts/Addition/apFWJBFB">addition</a>:
</p>

H
                  
                    2
                  
                
                (
                a
                ,
                b
                )
                =
                a
                ×
                b
                =
              
              
                
                  
                    
                      a
                      +
                      a
                      +
                      ⋯
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                      a
                    
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                b
                
                  
                     copies of 
                  
                
                a
              
            
          
        
      
    
    {\displaystyle {\begin{matrix}H_{2}(a,b)=a\times b=&\underbrace {a+a+\dots +a} \\&b{\mbox{ copies of }}a\end{matrix}}}

<p>For example,
</p>

4
                ×
                3
              
              
                =
              
              
                
                  
                    
                      4
                      +
                      4
                      +
                      4
                    
                    ⏟
                  
                
              
              
                =
              
              
                12
              
            
            
              
              
              
                3
                
                  
                     copies of 
                  
                
                4
              
            
          
        
      
    
    {\displaystyle {\begin{matrix}4\times 3&=&\underbrace {4+4+4} &=&12\\&&3{\mbox{ copies of }}4\end{matrix}}}

<p><a href="/facts/Exponentiation/pac6QY2g">Exponentiation</a> for a natural power 
  
    
      
        b
      
    
    {\displaystyle b}
  
 is defined as iterated multiplication, which Knuth denoted by a single up-arrow:
</p>

a
                ↑
                b
                =
                
                  H
                  
                    3
                  
                
                (
                a
                ,
                b
                )
                =
                
                  a
                  
                    b
                  
                
                =
              
              
                
                  
                    
                      a
                      ×
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                      ×
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                      a
                    
                    ⏟
                  
                
              
            
            
              
              
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                     copies of 
                  
                
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    {\displaystyle {\begin{matrix}a\uparrow b=H_{3}(a,b)=a^{b}=&\underbrace {a\times a\times \dots \times a} \\&b{\mbox{ copies of }}a\end{matrix}}}

<p>For example,
</p>

4
                ↑
                3
                =
                
                  4
                  
                    3
                  
                
                =
              
              
                
                  
                    
                      4
                      ×
                      4
                      ×
                      4
                    
                    ⏟
                  
                
              
              
                =
              
              
                64
              
            
            
              
              
                3
                
                  
                     copies of 
                  
                
                4
              
            
          
        
      
    
    {\displaystyle {\begin{matrix}4\uparrow 3=4^{3}=&\underbrace {4\times 4\times 4} &=&64\\&3{\mbox{ copies of }}4\end{matrix}}}

<p><a href="/facts/Tetration/5rKeVVYv">Tetration</a> is defined as iterated exponentiation, which Knuth denoted by a “double arrow”:
</p>

a
                ↑↑
                b
                =
                
                  H
                  
                    4
                  
                
                (
                a
                ,
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                )
                =
              
              
                
                  
                    
                      a
                      
                        
                          a

.
                                
                                  
                                  
                                    .
                                    
                                      
                                      
                                        .
                                        
                                          
                                          
                                            a
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
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                =
              
              
                
                  
                    
                      a
                      ↑
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                     copies of 
                  
                
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    {\displaystyle {\begin{matrix}a\uparrow \uparrow b=H_{4}(a,b)=&\underbrace {a^{a^{{}^{.\,^{.\,^{.\,^{a}}}}}}} &=&\underbrace {a\uparrow (a\uparrow (\dots \uparrow a))} \\&b{\mbox{ copies of }}a&&b{\mbox{ copies of }}a\end{matrix}}}

<p>For example,
</p>

4
                ↑↑
                3
                =
              
              
                
                  
                    
                      4
                      
                        
                          4
                          
                            4
                          
                        
                      
                    
                    ⏟
                  
                
              
              
                =
              
              
                
                  
                    
                      4
                      ↑
                      (
                      4
                      ↑
                      4
                      )
                    
                    ⏟
                  
                
              
              
                =
              
              
                
                  4
                  
                    256
                  
                
              
              
                ≈
              
              
                1.34078079
                ×
                
                  10
                  
                    154
                  
                
              
              
            
            
              
              
                3
                
                  
                     copies of 
                  
                
                4
              
              
              
                3
                
                  
                     copies of 
                  
                
                4
              
            
          
        
      
    
    {\displaystyle {\begin{matrix}4\uparrow \uparrow 3=&\underbrace {4^{4^{4}}} &=&\underbrace {4\uparrow (4\uparrow 4)} &=&4^{256}&\approx &1.34078079\times 10^{154}&\\&3{\mbox{ copies of }}4&&3{\mbox{ copies of }}4\end{matrix}}}

<p>Expressions are evaluated from right to left, as the operators are defined to be <a href="/facts/Right_associative_operator/HcEdArws">right-associative</a>.
</p><p>According to this definition,
</p>

3
        ↑↑
        2
        =
        
          3
          
            3
          
        
        =
        27
      
    
    {\displaystyle 3\uparrow \uparrow 2=3^{3}=27}

3
        ↑↑
        3
        =
        
          3
          
            
              3
              
                3
              
            
          
        
        =
        
          3
          
            27
          
        
        =
        7
        ,
        625
        ,
        597
        ,
        484
        ,
        987
      
    
    {\displaystyle 3\uparrow \uparrow 3=3^{3^{3}}=3^{27}=7,625,597,484,987}

3
        ↑↑
        4
        =
        
          3
          
            
              3
              
                
                  3
                  
                    3
                  
                
              
            
          
        
        =
        
          3
          
            
              3
              
                27
              
            
          
        
        =
        
          3
          
            7625597484987
          
        
        ≈
        1.2580143
        ×
        
          10
          
            3638334640024
          
        
      
    
    {\displaystyle 3\uparrow \uparrow 4=3^{3^{3^{3}}}=3^{3^{27}}=3^{7625597484987}\approx 1.2580143\times 10^{3638334640024}}

3
        ↑↑
        5
        =
        
          3
          
            
              3
              
                
                  3
                  
                    
                      3
                      
                        3
                      
                    
                  
                
              
            
          
        
        =
        
          3
          
            
              3
              
                
                  3
                  
                    27
                  
                
              
            
          
        
        =
        
          3
          
            
              3
              
                7625597484987
              
            
          
        
        ≈
        
          3
          
            1.2580143
            ×
            
              10
              
                3638334640024
              
            
          
        
      
    
    {\displaystyle 3\uparrow \uparrow 5=3^{3^{3^{3^{3}}}}=3^{3^{3^{27}}}=3^{3^{7625597484987}}\approx 3^{1.2580143\times 10^{3638334640024}}}

etc.
<p>This already leads to some fairly large numbers, but the hyperoperator sequence does not stop here.
</p><p><a href="/facts/Pentation/YKhnIWPG">Pentation</a>, defined as iterated tetration, is represented by the “triple arrow”:
</p>

a
                ↑↑↑
                b
                =
                
                  H
                  
                    5
                  
                
                (
                a
                ,
                b
                )
                =
              
              
                
                  
                    
                      
                        a

↑↑
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    {\displaystyle {\begin{matrix}a\uparrow \uparrow \uparrow b=H_{5}(a,b)=&\underbrace {a_{}\uparrow \uparrow (a\uparrow \uparrow (\dots \uparrow \uparrow a))} \\&b{\mbox{ copies of }}a\end{matrix}}}

<p><a href="/facts/Hexation/XNn1hneN">Hexation</a>, defined as iterated pentation, is represented by the “quadruple arrow”:
</p>

a
                ↑↑↑↑
                b
                =
                
                  H
                  
                    6
                  
                
                (
                a
                ,
                b
                )
                =
              
              
                
                  
                    
                      
                        a

↑↑↑
                      (
                      a
                      ↑↑↑
                      (
                      ⋯
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                      a
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                      )
                    
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                b
                
                  
                     copies of 
                  
                
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    {\displaystyle {\begin{matrix}a\uparrow \uparrow \uparrow \uparrow b=H_{6}(a,b)=&\underbrace {a_{}\uparrow \uparrow \uparrow (a\uparrow \uparrow \uparrow (\dots \uparrow \uparrow \uparrow a))} \\&b{\mbox{ copies of }}a\end{matrix}}}

<p>and so on. The general rule is that an 
  
    
      
        n
      
    
    {\displaystyle n}
  
-arrow operator expands into a right-associative series of (
  
    
      
        n
        −
        1
      
    
    {\displaystyle n-1}
  
)-arrow operators. Symbolically,
</p>

a
                 
                
                  
                    
                      
                        
                          ↑

↑
                        
                        
                        …
                        
                        
                        ↑
                      
                      ⏟
                    
                  
                  
                    n
                  
                
                 
                b
                =
                
                  
                    
                      
                        a
                         
                        
                          
                            
                              
                                ↑
                                
                                
                                …
                                
                                
                                ↑
                              
                              ⏟
                            
                          
                          
                            n
                            −
                            1
                          
                        
                         
                        (
                        a
                         
                        
                          
                            
                              
                                
                                  ↑

…
                                
                                
                                ↑
                              
                              ⏟
                            
                          
                          
                            n
                            −
                            1
                          
                        
                         
                        (
                        …
                         
                        
                          
                            
                              
                                
                                  ↑

…
                                
                                
                                ↑
                              
                              ⏟
                            
                          
                          
                            n
                            −
                            1
                          
                        
                         
                        a
                        )
                        )
                      
                      ⏟
                    
                  
                  
                    b
                    
                       copies of 
                    
                    a
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{matrix}a\ \underbrace {\uparrow _{}\uparrow \!\!\dots \!\!\uparrow } _{n}\ b=\underbrace {a\ \underbrace {\uparrow \!\!\dots \!\!\uparrow } _{n-1}\ (a\ \underbrace {\uparrow _{}\!\!\dots \!\!\uparrow } _{n-1}\ (\dots \ \underbrace {\uparrow _{}\!\!\dots \!\!\uparrow } _{n-1}\ a))} _{b{\text{ copies of }}a}\end{matrix}}}

<p>Examples:
</p>

3
        ↑↑↑
        2
        =
        3
        ↑↑
        3
        =
        
          3
          
            
              3
              
                3
              
            
          
        
        =
        
          3
          
            27
          
        
        =
        7
        ,
        625
        ,
        597
        ,
        484
        ,
        987
      
    
    {\displaystyle 3\uparrow \uparrow \uparrow 2=3\uparrow \uparrow 3=3^{3^{3}}=3^{27}=7,625,597,484,987}

3
                ↑↑↑
                3
                =
                3
                ↑↑
                (
                3
                ↑↑
                3
                )
                =
                3
                ↑↑
                (
                3
                ↑
                3
                ↑
                3
                )
                =
              
              
                
                  
                    
                      
                        3

↑
                      3
                      ↑
                      ⋯
                      ↑
                      3
                    
                    ⏟
                  
                
              
            
            
              
              
                3
                ↑
                3
                ↑
                3
                
                  
                     copies of 
                  
                
                3
              
            
          
        
        
          
            
              
                =
              
              
                
                  
                    
                      
                        3

↑
                      3
                      ↑
                      ⋯
                      ↑
                      3
                    
                    ⏟
                  
                
              
            
            
              
              
                
                  
                    7,625,597,484,987 copies of 3
                  
                
              
            
          
        
        
          
            
              
                =
              
              
                
                  
                    
                      3
                      
                        
                          3
                          
                            
                              3
                              
                                
                                  3
                                  
                                    
                                      ⋅
                                      
                                        
                                          ⋅
                                          
                                            
                                              ⋅
                                              
                                                
                                                  ⋅
                                                  
                                                    3
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    ⏟
                  
                
              
            
            
              
              
                
                  
                    7,625,597,484,987 copies of 3
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{matrix}3\uparrow \uparrow \uparrow 3=3\uparrow \uparrow (3\uparrow \uparrow 3)=3\uparrow \uparrow (3\uparrow 3\uparrow 3)=&\underbrace {3_{}\uparrow 3\uparrow \dots \uparrow 3} \\&3\uparrow 3\uparrow 3{\mbox{ copies of }}3\end{matrix}}{\begin{matrix}=&\underbrace {3_{}\uparrow 3\uparrow \dots \uparrow 3} \\&{\mbox{7,625,597,484,987 copies of 3}}\end{matrix}}{\begin{matrix}=&\underbrace {3^{3^{3^{3^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{3}}}}}}}}} \\&{\mbox{7,625,597,484,987 copies of 3}}\end{matrix}}}

<h2 id="notation">Notation</h2>
<p>In expressions such as 
  
    
      
        
          a
          
            b
          
        
      
    
    {\displaystyle a^{b}}
  
, the notation for exponentiation is usually to write the exponent 
  
    
      
        b
      
    
    {\displaystyle b}
  
 as a superscript to the base number 
  
    
      
        a
      
    
    {\displaystyle a}
  
. But many environments — such as <a href="/facts/Programming_language/cWTYbgWa">programming languages</a> and plain-text <a href="/facts/E-mail/6pD9GLtI">e-mail</a> — do not support <a href="/facts/Superscript/6tExBnkw">superscript</a> typesetting. People have adopted the linear notation 
  
    
      
        a
        ↑
        b
      
    
    {\displaystyle a\uparrow b}
  
 for such environments; the up-arrow suggests 'raising to the power of'. If the <a href="/facts/Character_set/ShJHIMoA">character set</a> does not contain an up arrow, the <a href="/facts/Caret/LRV5MF7s">caret</a> (^) is used instead.
</p><p>The superscript notation 
  
    
      
        
          a
          
            b
          
        
      
    
    {\displaystyle a^{b}}
  
 doesn't lend itself well to generalization, which explains why Knuth chose to work from the inline notation 
  
    
      
        a
        ↑
        b
      
    
    {\displaystyle a\uparrow b}
  
 instead.
</p><p>
  
    
      
        a
        
          ↑
          
            n
          
        
        b
      
    
    {\displaystyle a\uparrow ^{n}b}
  
 is a shorter alternative notation for n uparrows. Thus 
  
    
      
        a
        
          ↑
          
            4
          
        
        b
        =
        a
        ↑↑↑↑
        b
      
    
    {\displaystyle a\uparrow ^{4}b=a\uparrow \uparrow \uparrow \uparrow b}
  
.
</p>
<h3>Writing out up-arrow notation in terms of powers</h3>
<p>Attempting to write 
  
    
      
        a
        ↑↑
        b
      
    
    {\displaystyle a\uparrow \uparrow b}
  
 using the familiar superscript notation gives a <a href="/facts/Tetration/5rKeVVYv">power tower</a>.
</p>
For example: 
  
    
      
        a
        ↑↑
        4
        =
        a
        ↑
        (
        a
        ↑
        (
        a
        ↑
        a
        )
        )
        =
        
          a
          
            
              a
              
                
                  a
                  
                    a
                  
                
              
            
          
        
      
    
    {\displaystyle a\uparrow \uparrow 4=a\uparrow (a\uparrow (a\uparrow a))=a^{a^{a^{a}}}}

<p>If 
  
    
      
        b
      
    
    {\displaystyle b}
  
 is a variable (or is too large), the power tower might be written using dots and a note indicating the height of the tower.
</p>

a
        ↑↑
        b
        =
        
          
            
              
                a
                
                  
                    a
                    
                      
                        .
                        
                          
                            .
                            
                              .
                              
                                a
                              
                            
                          
                        
                      
                    
                  
                
              
              ⏟
            
          
          
            b
          
        
      
    
    {\displaystyle a\uparrow \uparrow b=\underbrace {a^{a^{.^{.^{.{a}}}}}} _{b}}

<p>Continuing with this notation, 
  
    
      
        a
        ↑↑↑
        b
      
    
    {\displaystyle a\uparrow \uparrow \uparrow b}
  
 could be written with a stack of such power towers, each describing the size of the one above it.
</p>

a
        ↑↑↑
        4
        =
        a
        ↑↑
        (
        a
        ↑↑
        (
        a
        ↑↑
        a
        )
        )
        =
        
          
            
              
                a
                
                  
                    a
                    
                      
                        .
                        
                          
                            .
                            
                              .
                              
                                a
                              
                            
                          
                        
                      
                    
                  
                
              
              ⏟
            
          
          
            
              
                
                  
                    a
                    
                      
                        a
                        
                          
                            .
                            
                              
                                .
                                
                                  .
                                  
                                    a
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  ⏟
                
              
              
                
                  
                    
                      
                        a
                        
                          
                            a
                            
                              
                                .
                                
                                  
                                    .
                                    
                                      .
                                      
                                        a
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ⏟
                    
                  
                  
                    a
                  
                
              
            
          
        
      
    
    {\displaystyle a\uparrow \uparrow \uparrow 4=a\uparrow \uparrow (a\uparrow \uparrow (a\uparrow \uparrow a))=\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{a}}}}

<p>Again, if 
  
    
      
        b
      
    
    {\displaystyle b}
  
 is a variable or is too large, the stack might be written using dots and a note indicating its height.
</p>

a
        ↑↑↑
        b
        =
        
          
          
            
              
                
                  a
                  
                    
                      a
                      
                        
                          .
                          
                            
                              .
                              
                                .
                                
                                  a
                                
                              
                            
                          
                        
                      
                    
                  
                
                ⏟
              
            
            
              
                
                  
                    
                      a
                      
                        
                          a
                          
                            
                              .
                              
                                
                                  .
                                  
                                    .
                                    
                                      a
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    ⏟
                  
                
                
                  
                    
                      
                        ⋮
                        ⏟
                      
                    
                    
                      a
                    
                  
                
              
            
          
          }
        
        b
      
    
    {\displaystyle a\uparrow \uparrow \uparrow b=\left.\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {\vdots } _{a}}}\right\}b}

<p>Furthermore, 
  
    
      
        a
        ↑↑↑↑
        b
      
    
    {\displaystyle a\uparrow \uparrow \uparrow \uparrow b}
  
 might be written using several columns of such stacks of power towers, each column describing the number of power towers in the stack to its left:
</p>

a
        ↑↑↑↑
        4
        =
        a
        ↑↑↑
        (
        a
        ↑↑↑
        (
        a
        ↑↑↑
        a
        )
        )
        =
        
          
          
            
              
              
                
                  
                  
                    
                      
                        
                          a
                          
                            
                              a
                              
                                
                                  .
                                  
                                    
                                      .
                                      
                                        .
                                        
                                          a
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        ⏟
                      
                    
                    
                      
                        
                          
                            
                              a
                              
                                
                                  a
                                  
                                    
                                      .
                                      
                                        
                                          .
                                          
                                            .
                                            
                                              a
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                            ⏟
                          
                        
                        
                          
                            
                              
                                ⋮
                                ⏟
                              
                            
                            
                              a
                            
                          
                        
                      
                    
                  
                  }
                
                
                  
                    
                      
                        a
                        
                          
                            a
                            
                              
                                .
                                
                                  
                                    .
                                    
                                      .
                                      
                                        a
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ⏟
                    
                  
                  
                    
                      
                        
                          
                            a
                            
                              
                                a
                                
                                  
                                    .
                                    
                                      
                                        .
                                        
                                          .
                                          
                                            a
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                          ⏟
                        
                      
                      
                        
                          
                            
                              ⋮
                              ⏟
                            
                          
                          
                            a
                          
                        
                      
                    
                  
                
              
              }
            
            
              
                
                  
                    a
                    
                      
                        a
                        
                          
                            .
                            
                              
                                .
                                
                                  .
                                  
                                    a
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                  ⏟
                
              
              
                
                  
                    
                      
                        a
                        
                          
                            a
                            
                              
                                .
                                
                                  
                                    .
                                    
                                      .
                                      
                                        a
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                      ⏟
                    
                  
                  
                    
                      
                        
                          ⋮
                          ⏟
                        
                      
                      
                        a
                      
                    
                  
                
              
            
          
          }
        
        a
      
    
    {\displaystyle a\uparrow \uparrow \uparrow \uparrow 4=a\uparrow \uparrow \uparrow (a\uparrow \uparrow \uparrow (a\uparrow \uparrow \uparrow a))=\left.\left.\left.\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {\vdots } _{a}}}\right\}\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {\vdots } _{a}}}\right\}\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {\vdots } _{a}}}\right\}a}

<p>And more generally:
</p>

a
        ↑↑↑↑
        b
        =
        
          
            
              
                
                  
                  
                    
                      
                      
                        
                          
                          
                            
                              
                                
                                  a
                                  
                                    
                                      a
                                      
                                        
                                          .
                                          
                                            
                                              .
                                              
                                                .
                                                
                                                  a
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                                ⏟
                              
                            
                            
                              
                                
                                  
                                    
                                      a
                                      
                                        
                                          a
                                          
                                            
                                              .
                                              
                                                
                                                  .
                                                  
                                                    .
                                                    
                                                      a
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                    ⏟
                                  
                                
                                
                                  
                                    
                                      
                                        ⋮
                                        ⏟
                                      
                                    
                                    
                                      a
                                    
                                  
                                
                              
                            
                          
                          }
                        
                        
                          
                            
                              
                                a
                                
                                  
                                    a
                                    
                                      
                                        .
                                        
                                          
                                            .
                                            
                                              .
                                              
                                                a
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                              ⏟
                            
                          
                          
                            
                              
                                
                                  
                                    a
                                    
                                      
                                        a
                                        
                                          
                                            .
                                            
                                              
                                                .
                                                
                                                  .
                                                  
                                                    a
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                  ⏟
                                
                              
                              
                                
                                  
                                    
                                      ⋮
                                      ⏟
                                    
                                  
                                  
                                    a
                                  
                                
                              
                            
                          
                        
                      
                      }
                    
                    ⋯
                  
                  }
                
                a
              
              ⏟
            
          
          
            b
          
        
      
    
    {\displaystyle a\uparrow \uparrow \uparrow \uparrow b=\underbrace {\left.\left.\left.\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {\vdots } _{a}}}\right\}\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {\vdots } _{a}}}\right\}\cdots \right\}a} _{b}}

<p>This might be carried out indefinitely to represent 
  
    
      
        a
        
          ↑
          
            n
          
        
        b
      
    
    {\displaystyle a\uparrow ^{n}b}
  
 as iterated exponentiation of iterated exponentiation for any 
  
    
      
        a
      
    
    {\displaystyle a}
  
, 
  
    
      
        n
      
    
    {\displaystyle n}
  
, and 
  
    
      
        b
      
    
    {\displaystyle b}
  
 (although it clearly becomes rather cumbersome).
</p>
<h4>Using tetration</h4>
<p>The Rudy Rucker notation 
  
    
      
        
          
          
            b
          
        
        a
      
    
    {\displaystyle ^{b}a}
  
 for <a href="/facts/Tetration/5rKeVVYv">tetration</a> allows us to make these diagrams slightly simpler while still employing a geometric representation (we could call these <i>tetration towers</i>).
</p>

a
        ↑↑
        b
        =

b
          
        
        a
      
    
    {\displaystyle a\uparrow \uparrow b={}^{b}a}

a
        ↑↑↑
        b
        =
        
          
            
              
                
                  
                  
                    
                      
                      
                        
                          
                          
                            
                              
                              
                                
                                  
                                  
                                    a
                                  
                                
                                .
                              
                            
                            .
                          
                        
                        .
                      
                    
                    a
                  
                
                a
              
              ⏟
            
          
          
            b
          
        
      
    
    {\displaystyle a\uparrow \uparrow \uparrow b=\underbrace {^{^{^{^{^{a}.}.}.}a}a} _{b}}

a
        ↑↑↑↑
        b
        =
        
          
          
            
              
                
                  
                    
                    
                      
                        
                        
                          
                            
                            
                              
                                
                                
                                  
                                    
                                    
                                      a
                                    
                                  
                                  .
                                
                              
                              .
                            
                          
                          .
                        
                      
                      a
                    
                  
                  a
                
                ⏟
              
            
            
              
                
                  
                    
                      
                        
                        
                          
                            
                            
                              
                                
                                
                                  
                                    
                                    
                                      
                                        
                                        
                                          a
                                        
                                      
                                      .
                                    
                                  
                                  .
                                
                              
                              .
                            
                          
                          a
                        
                      
                      a
                    
                    ⏟
                  
                
                
                  
                    
                      
                        ⋮
                        ⏟
                      
                    
                    
                      a
                    
                  
                
              
            
          
          }
        
        b
      
    
    {\displaystyle a\uparrow \uparrow \uparrow \uparrow b=\left.\underbrace {^{^{^{^{^{a}.}.}.}a}a} _{\underbrace {^{^{^{^{^{a}.}.}.}a}a} _{\underbrace {\vdots } _{a}}}\right\}b}

<p>Finally, as an example, the fourth Ackermann number 
  
    
      
        4
        
          ↑
          
            4
          
        
        4
      
    
    {\displaystyle 4\uparrow ^{4}4}
  
 could be represented as:
</p>

4
                                  
                                
                                .
                              
                            
                            .
                          
                        
                        .
                      
                    
                    4
                  
                
                4
              
              ⏟
            
          
          
            
              
                
                  
                    
                      
                      
                        
                          
                          
                            
                              
                              
                                
                                  
                                  
                                    
                                      
                                      
                                        4
                                      
                                    
                                    .
                                  
                                
                                .
                              
                            
                            .
                          
                        
                        4
                      
                    
                    4
                  
                  ⏟
                
              
              
                
                  
                    
                      
                        
                          
                          
                            
                              
                              
                                
                                  
                                  
                                    
                                      
                                      
                                        
                                          
                                          
                                            4
                                          
                                        
                                        .
                                      
                                    
                                    .
                                  
                                
                                .
                              
                            
                            4
                          
                        
                        4
                      
                      ⏟
                    
                  
                  
                    4
                  
                
              
            
          
        
        =
        
          
            
              
                
                  
                  
                    
                      
                      
                        
                          
                          
                            
                              
                              
                                
                                  
                                  
                                    4
                                  
                                
                                .
                              
                            
                            .
                          
                        
                        .
                      
                    
                    4
                  
                
                4
              
              ⏟
            
          
          
            
              
                
                  
                    
                      
                      
                        
                          
                          
                            
                              
                              
                                
                                  
                                  
                                    
                                      
                                      
                                        4
                                      
                                    
                                    .
                                  
                                
                                .
                              
                            
                            .
                          
                        
                        4
                      
                    
                    4
                  
                  ⏟
                
              
              
                
                  
                  
                    
                      
                      
                        
                          
                          
                            4
                          
                        
                        4
                      
                    
                    4
                  
                
                4
              
            
          
        
      
    
    {\displaystyle \underbrace {^{^{^{^{^{4}.}.}.}4}4} _{\underbrace {^{^{^{^{^{4}.}.}.}4}4} _{\underbrace {^{^{^{^{^{4}.}.}.}4}4} _{4}}}=\underbrace {^{^{^{^{^{4}.}.}.}4}4} _{\underbrace {^{^{^{^{^{4}.}.}.}4}4} _{^{^{^{4}4}4}4}}}

<h2 id="generalizations">Generalizations</h2>
<p>Some numbers are so large that multiple arrows of Knuth's up-arrow notation become too cumbersome; then an <i>n</i>-arrow operator  
  
    
      
        
          ↑
          
            n
          
        
      
    
    {\displaystyle \uparrow ^{n}}
  
 is useful (and also for descriptions with a variable number of arrows), or equivalently, <a href="/facts/Hyper_operator/XNn1hneN">hyper operators</a>.
</p><p>Some numbers are so large that even that notation is not sufficient. The <a href="/facts/Conway_chained_arrow_notation/VHlJgdd8">Conway chained arrow notation</a> can then be used: a chain of three elements is equivalent with the other notations, but a chain of four or more is even more powerful.
</p>

a
                
                  ↑
                  
                    n
                  
                
                b
              
              
                =
              
              
                a
                [
                n
                +
                2
                ]
                b
              
              
                =
              
              
                a
                →
                b
                →
                n
              
            
            
              
                
                  
                    (Knuth)
                  
                
              
              
              
                
                  
                    (hyperoperation)
                  
                
              
              
              
                
                  
                    (Conway)
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{matrix}a\uparrow ^{n}b&=&a[n+2]b&=&a\to b\to n\\{\mbox{(Knuth)}}&&{\mbox{(hyperoperation)}}&&{\mbox{(Conway)}}\end{matrix}}}

<p>
  
    
      
        6
        ↑↑
        4
      
    
    {\displaystyle 6\uparrow \uparrow 4}
  
 = 
  
    
      
        
          
            
              
                6
                
                  
                    6
                    
                      
                        .
                        
                          
                            .
                            
                              
                                .
                                
                                  6
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              ⏟
            
          
          
            4
          
        
      
    
    {\displaystyle \underbrace {6^{6^{.^{.^{.^{6}}}}}} _{4}}
  
, Since 
  
    
      
        6
        ↑↑
        4
      
    
    {\displaystyle 6\uparrow \uparrow 4}
  
 = 
  
    
      
        
          6
          
            
              6
              
                
                  6
                  
                    6
                  
                
              
            
          
        
      
    
    {\displaystyle 6^{6^{6^{6}}}}
  
 = 
  
    
      
        
          6
          
            
              6
              
                46
                ,
                656
              
            
          
        
      
    
    {\displaystyle 6^{6^{46,656}}}
  
, Thus the result comes out with 
  
    
      
        
          
            
              
                6
                
                  
                    6
                    
                      
                        .
                        
                          
                            .
                            
                              
                                .
                                
                                  6
                                
                              
                            
                          
                        
                      
                    
                  
                
              
              ⏟
            
          
          
            4
          
        
      
    
    {\displaystyle \underbrace {6^{6^{.^{.^{.^{6}}}}}} _{4}}

</p><p>
  
    
      
        10
        ↑
        (
        3
        ×
        10
        ↑
        (
        3
        ×
        10
        ↑
        15
        )
        +
        3
        )
      
    
    {\displaystyle 10\uparrow (3\times 10\uparrow (3\times 10\uparrow 15)+3)}
  
 = 
  
    
      
        
          
            
              100000...000
              ⏟
            
          
          
            
              
                
                  300000...003
                  ⏟
                
              
              
                
                  
                    
                      300000...000
                      ⏟
                    
                  
                  
                    15
                  
                
              
            
          
        
      
    
    {\displaystyle \underbrace {100000...000} _{\underbrace {300000...003} _{\underbrace {300000...000} _{15}}}}
  
 or 
  
    
      
        
          10
          
            3
            ×
            
              10
              
                3
                ×
                
                  10
                  
                    15
                  
                
              
            
            +
            3
          
        
      
    
    {\displaystyle 10^{3\times 10^{3\times 10^{15}}+3}}

</p><p>Even faster-growing functions can be categorized using an <a href="/facts/Ordinal_number/GelFLjJt">ordinal</a> analysis called the <a href="/facts/Fast-growing_hierarchy/4ctDqlLY">fast-growing hierarchy</a>. The fast-growing hierarchy uses successive function iteration and diagonalization to systematically create faster-growing functions from some base function 
  
    
      
        f
        (
        x
        )
      
    
    {\displaystyle f(x)}
  
. For the standard fast-growing hierarchy using 
  
    
      
        
          f
          
            0
          
        
        (
        x
        )
        =
        x
        +
        1
      
    
    {\displaystyle f_{0}(x)=x+1}
  
, 
  
    
      
        
          f
          
            2
          
        
        (
        x
        )
      
    
    {\displaystyle f_{2}(x)}
  
 already exhibits exponential growth, 
  
    
      
        
          f
          
            3
          
        
        (
        x
        )
      
    
    {\displaystyle f_{3}(x)}
  
 is comparable to tetrational growth and is upper-bounded by a function involving the first four hyperoperators;. Then, 
  
    
      
        
          f
          
            ω
          
        
        (
        x
        )
      
    
    {\displaystyle f_{\omega }(x)}
  
 is comparable to the <a href="/facts/Ackermann_function/WQNpCWHn">Ackermann function</a>, 
  
    
      
        
          f
          
            ω
            +
            1
          
        
        (
        x
        )
      
    
    {\displaystyle f_{\omega +1}(x)}
  
 is already beyond the reach of indexed arrows but can be used to approximate <a href="/facts/Graham%27s_number/7IJYxN2x">Graham's number</a>, and 
  
    
      
        
          f
          
            
              ω
              
                2
              
            
          
        
        (
        x
        )
      
    
    {\displaystyle f_{\omega ^{2}}(x)}
  
 is comparable to arbitrarily-long Conway chained arrow notation.
</p><p>These functions are all computable. Even faster computable functions, such as the <a href="/facts/Goodstein_sequence/W2SvUUpk">Goodstein sequence</a> and the <a href="/facts/TREE(3)/3q0G4cbP">TREE sequence</a> require the usage of large ordinals, may occur in certain combinatorical and proof-theoretic contexts. There exist functions which grow uncomputably fast, such as the <a href="/facts/Busy_Beaver/qfdJPkVu">Busy Beaver</a>, whose very nature will be completely out of reach from any up-arrow, or even any ordinal-based analysis.
</p>
<h2 id="definition">Definition</h2>
<p>Without reference to <a href="/facts/Hyperoperation/XNn1hneN">hyperoperation</a> the up-arrow operators can be formally defined by
</p>

a
        
          ↑
          
            n
          
        
        b
        =
        
          
            {
            
              
                
                  
                    a
                    
                      b
                    
                  
                  ,
                
                
                  
                    if 
                  
                  n
                  =
                  1
                  ;
                
              
              
                
                  1
                  ,
                
                
                  
                    if 
                  
                  n
                  >
                  1
                  
                     and 
                  
                  b
                  =
                  0
                  ;
                
              
              
                
                  a
                  
                    ↑
                    
                      n
                      −
                      1
                    
                  
                  (
                  a
                  
                    ↑
                    
                      n
                    
                  
                  (
                  b
                  −
                  1
                  )
                  )
                  ,
                
                
                  
                    otherwise 
                  
                
              
            
            
          
        
      
    
    {\displaystyle a\uparrow ^{n}b={\begin{cases}a^{b},&{\text{if }}n=1;\\1,&{\text{if }}n>1{\text{ and }}b=0;\\a\uparrow ^{n-1}(a\uparrow ^{n}(b-1)),&{\text{otherwise }}\end{cases}}}

<p>for all integers 
  
    
      
        a
        ,
        b
        ,
        n
      
    
    {\displaystyle a,b,n}
  
 with 
  
    
      
        a
        ≥
        0
        ,
        n
        ≥
        1
        ,
        b
        ≥
        0
      
    
    {\displaystyle a\geq 0,n\geq 1,b\geq 0}
  
<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>.
</p><p>This definition uses <a href="/facts/Exponentiation/pac6QY2g">exponentiation</a> 
  
    
      
        (
        a
        
          ↑
          
            1
          
        
        b
        =
        a
        ↑
        b
        =
        
          a
          
            b
          
        
        )
      
    
    {\displaystyle (a\uparrow ^{1}b=a\uparrow b=a^{b})}
  
 as the base case, and <a href="/facts/Tetration/5rKeVVYv">tetration</a> 
  
    
      
        (
        a
        
          ↑
          
            2
          
        
        b
        =
        a
        ↑↑
        b
        )
      
    
    {\displaystyle (a\uparrow ^{2}b=a\uparrow \uparrow b)}
  
 as repeated exponentiation.  This is equivalent to the <a href="/facts/Hyperoperation/XNn1hneN">hyperoperation sequence</a> except it omits the three more basic operations of <a href="/facts/Successor_function/R1hhkSEG">succession</a>, <a href="/facts/Addition/apFWJBFB">addition</a> and <a href="/facts/Multiplication/VtcUjpNv">multiplication</a>.
</p><p>One can alternatively choose <a href="/facts/Multiplication/VtcUjpNv">multiplication</a> 
  
    
      
        (
        a
        
          ↑
          
            0
          
        
        b
        =
        a
        ×
        b
        )
      
    
    {\displaystyle (a\uparrow ^{0}b=a\times b)}
  
 as the base case and iterate from there. Then <a href="/facts/Exponentiation/pac6QY2g">exponentiation</a> becomes repeated multiplication. The formal definition would be
</p>

a
        
          ↑
          
            n
          
        
        b
        =
        
          
            {
            
              
                
                  a
                  ×
                  b
                  ,
                
                
                  
                    if 
                  
                  n
                  =
                  0
                  ;
                
              
              
                
                  1
                  ,
                
                
                  
                    if 
                  
                  n
                  >
                  0
                  
                     and 
                  
                  b
                  =
                  0
                  ;
                
              
              
                
                  a
                  
                    ↑
                    
                      n
                      −
                      1
                    
                  
                  (
                  a
                  
                    ↑
                    
                      n
                    
                  
                  (
                  b
                  −
                  1
                  )
                  )
                  ,
                
                
                  
                    otherwise 
                  
                
              
            
            
          
        
      
    
    {\displaystyle a\uparrow ^{n}b={\begin{cases}a\times b,&{\text{if }}n=0;\\1,&{\text{if }}n>0{\text{ and }}b=0;\\a\uparrow ^{n-1}(a\uparrow ^{n}(b-1)),&{\text{otherwise }}\end{cases}}}

<p>for all integers 
  
    
      
        a
        ,
        b
        ,
        n
      
    
    {\displaystyle a,b,n}
  
 with 
  
    
      
        a
        ≥
        0
        ,
        n
        ≥
        0
        ,
        b
        ≥
        0
      
    
    {\displaystyle a\geq 0,n\geq 0,b\geq 0}
  
.
</p><p>Note, however, that Knuth did not define the "nil-arrow" (
  
    
      
        
          ↑
          
            0
          
        
      
    
    {\displaystyle \uparrow ^{0}}
  
). One could extend the notation to negative indices (n ≥ -2) in such a way as to agree with the entire hyperoperation sequence, except for the lag in the indexing:
</p>

H
          
            n
          
        
        (
        a
        ,
        b
        )
        =
        a
        [
        n
        ]
        b
        =
        a
        
          ↑
          
            n
            −
            2
          
        
        b
        
           for 
        
        n
        ≥
        0.
      
    
    {\displaystyle H_{n}(a,b)=a[n]b=a\uparrow ^{n-2}b{\text{ for }}n\geq 0.}

<p>The up-arrow operation is a <a href="/facts/Associative_property/EQX8lV6r">right-associative operation</a>, that is, 
  
    
      
        a
        ↑
        b
        ↑
        c
      
    
    {\displaystyle a\uparrow b\uparrow c}
  
 is understood to be 
  
    
      
        a
        ↑
        (
        b
        ↑
        c
        )
      
    
    {\displaystyle a\uparrow (b\uparrow c)}
  
, instead of 
  
    
      
        (
        a
        ↑
        b
        )
        ↑
        c
      
    
    {\displaystyle (a\uparrow b)\uparrow c}
  
. If ambiguity is not an issue parentheses are sometimes dropped.
</p>
<h2 id="tables-of-values">Tables of values</h2>
<h3>Computing 0↑<i>n</i> <i>b</i></h3>
<p>Computing 
  
    
      
        0
        
          ↑
          
            n
          
        
        b
        =
        
          H
          
            n
            +
            2
          
        
        (
        0
        ,
        b
        )
        =
        0
        [
        n
        +
        2
        ]
        b
      
    
    {\displaystyle 0\uparrow ^{n}b=H_{n+2}(0,b)=0[n+2]b}
  
 results in
</p>
0, when <i>n</i> = 0  <a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
1, when <i>n</i> = 1 and <i>b</i> = 0   <a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
0, when <i>n</i> = 1 and <i>b</i> > 0   <a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>
1, when <i>n</i> > 1 and <i>b</i> is even (including 0)
0, when <i>n</i> > 1 and <i>b</i> is odd
<h3>Computing 2↑<i>n</i> <i>b</i></h3>
<p>Computing 
  
    
      
        2
        
          ↑
          
            n
          
        
        b
      
    
    {\displaystyle 2\uparrow ^{n}b}
  
 can be restated in terms of an infinite table. We place the numbers 
  
    
      
        
          2
          
            b
          
        
      
    
    {\displaystyle 2^{b}}
  
 in the top row, and fill the left column with values 2. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.
</p>
Values of                     2                  ↑                      n                          b              {\displaystyle 2\uparrow ^{n}b}   = <a href="/facts/Hyperoperation/XNn1hneN">                              H                      n            +            2                          (        2        ,        b        )              {\displaystyle H_{n+2}(2,b)}  </a> = <a href="/facts/Hyperoperation/XNn1hneN">                    2        [        n        +        2        ]        b              {\displaystyle 2[n+2]b}  </a> = <a href="/facts/Conway_chained_arrow_notation/VHlJgdd8">2 → b → n</a><table><tbody><tr><th><i>b</i><i>ⁿ</i></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>formula</th></tr><tr><th>1</th><td>2</td><td>4</td><td>8</td><td>16</td><td>32</td><td>64</td><td>                              2                      b                                {\displaystyle 2^{b}}  </td></tr><tr><th>2</th><td>2</td><td>4</td><td>16</td><td>65536</td><td>                              2                      65,536                          ≈        2.0        ×                  10                      19,728                                {\displaystyle 2^{65{,}536}\approx 2.0\times 10^{19{,}728}}  </td><td>                              2                                    2                              65,536                                                    ≈                  10                      6.0            ×                          10                              19,727                                                          {\displaystyle 2^{2^{65{,}536}}\approx 10^{6.0\times 10^{19{,}727}}}  </td><td>                    2        ↑↑        b              {\displaystyle 2\uparrow \uparrow b}  </td></tr><tr><th>3</th><td>2</td><td>4</td><td>65536</td><td>                                                                                                                                    2                                                                                                                    2                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        2                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                      65,536                                                       copies of                                                   2                                                          {\displaystyle {\begin{matrix}\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\65{,}536{\mbox{ copies of }}2\end{matrix}}}  </td><td>                                                                                                                                    2                                                                                                                    2                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        2                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  2                                                                                                                    2                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        2                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                      65,536                                                       copies of                                                   2                                                          {\displaystyle {\begin{matrix}\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\65{,}536{\mbox{ copies of }}2\end{matrix}}}  </td><td>                                                                                                                                    2                                                                                                                    2                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        2                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  2                                                                                                                    2                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        2                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  2                                                                                                                    2                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        2                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                      65,536                                                       copies of                                                   2                                                          {\displaystyle {\begin{matrix}\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\65{,}536{\mbox{ copies of }}2\end{matrix}}}  </td><td>                    2        ↑↑↑        b              {\displaystyle 2\uparrow \uparrow \uparrow b}  </td></tr><tr><th>4</th><td>2</td><td>4</td><td>                                                                                                                                    2                                                                                                                    2                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        2                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                      65,536                                                       copies of                                                   2                                                          {\displaystyle {\begin{matrix}\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\65{,}536{\mbox{ copies of }}2\end{matrix}}}  </td><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              2                                                                                                                    .                                                                                                        .                                                                                            .                                                                                2                                                                    2                                        ⏟                                                                                                                                                                  2                                                                                                                    2                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        2                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                      65,536                                                       copies of                                                   2                                                          {\displaystyle {\begin{matrix}\underbrace {^{^{^{^{^{2}.}.}.}2}2} \\\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\65{,}536{\mbox{ copies of }}2\end{matrix}}}  </td><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              2                                                                                                                    .                                                                                                        .                                                                                            .                                                                                2                                                                    2                                        ⏟                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            2                                                                                                                    .                                                                                                        .                                                                                            .                                                                                2                                                                    2                                        ⏟                                                                                                                                                                  2                                                                                                                    2                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        2                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                      65,536                                                       copies of                                                   2                                                          {\displaystyle {\begin{matrix}\underbrace {^{^{^{^{^{2}.}.}.}2}2} \\\underbrace {^{^{^{^{^{2}.}.}.}2}2} \\\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\65{,}536{\mbox{ copies of }}2\end{matrix}}}  </td><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              2                                                                                                                    .                                                                                                        .                                                                                            .                                                                                2                                                                    2                                        ⏟                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            2                                                                                                                    .                                                                                                        .                                                                                            .                                                                                2                                                                    2                                        ⏟                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            2                                                                                                                    .                                                                                                        .                                                                                            .                                                                                2                                                                    2                                        ⏟                                                                                                                                                                  2                                                                                                                    2                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        2                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                      65,536                                                       copies of                                                   2                                                          {\displaystyle {\begin{matrix}\underbrace {^{^{^{^{^{2}.}.}.}2}2} \\\underbrace {^{^{^{^{^{2}.}.}.}2}2} \\\underbrace {^{^{^{^{^{2}.}.}.}2}2} \\\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\65{,}536{\mbox{ copies of }}2\end{matrix}}}  </td><td>                    2        ↑↑↑↑        b              {\displaystyle 2\uparrow \uparrow \uparrow \uparrow b}  </td></tr></tbody></table>
<p>The table is the same as <a href="/facts/Ackermann_function/WQNpCWHn">that of the Ackermann function</a>, except for a shift in 
  
    
      
        n
      
    
    {\displaystyle n}
  
 and 
  
    
      
        b
      
    
    {\displaystyle b}
  
, and an addition of 3 to all values.
</p>
<h3>Computing 3↑<i>n</i> <i>b</i></h3>
<p>We place the numbers 
  
    
      
        
          3
          
            b
          
        
      
    
    {\displaystyle 3^{b}}
  
 in the top row, and fill the left column with values 3. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.
</p>
Values of                     3                  ↑                      n                          b              {\displaystyle 3\uparrow ^{n}b}   = <a href="/facts/Hyperoperation/XNn1hneN">                              H                      n            +            2                          (        3        ,        b        )              {\displaystyle H_{n+2}(3,b)}  </a> = <a href="/facts/Hyperoperation/XNn1hneN">                    3        [        n        +        2        ]        b              {\displaystyle 3[n+2]b}  </a> = <a href="/facts/Conway_chained_arrow_notation/VHlJgdd8">3 → b → n</a><table><tbody><tr><th><i>b</i><i>ⁿ</i></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>formula</th></tr><tr><th>1</th><td>3</td><td>9</td><td>27</td><td>81</td><td>243</td><td>                              3                      b                                {\displaystyle 3^{b}}  </td></tr><tr><th>2</th><td>3</td><td>27</td><td>7,625,597,484,987</td><td>                              3                      7,625,597,484,987                          ≈        1.3        ×                  10                      3,638,334,640,024                                {\displaystyle 3^{7{,}625{,}597{,}484{,}987}\approx 1.3\times 10^{3{,}638{,}334{,}640{,}024}}  </td><td>                              3                                    3                              7,625,597,484,987                                                          {\displaystyle 3^{3^{7{,}625{,}597{,}484{,}987}}}  </td><td>                    3        ↑↑        b              {\displaystyle 3\uparrow \uparrow b}  </td></tr><tr><th>3</th><td>3</td><td>7,625,597,484,987</td><td>                                                                                                                                    3                                                                                                                    3                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        3                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                      7,625,597,484,987                                                       copies of                                                   3                                                          {\displaystyle {\begin{matrix}\underbrace {3_{}^{3^{{}^{.\,^{.\,^{.\,^{3}}}}}}} \\7{,}625{,}597{,}484{,}987{\mbox{ copies of }}3\end{matrix}}}  </td><td>                                                                                                                                    3                                                                                                                    3                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        3                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  3                                                                                                                    3                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        3                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                      7,625,597,484,987                                                       copies of                                                   3                                                          {\displaystyle {\begin{matrix}\underbrace {3_{}^{3^{{}^{.\,^{.\,^{.\,^{3}}}}}}} \\\underbrace {3_{}^{3^{{}^{.\,^{.\,^{.\,^{3}}}}}}} \\7{,}625{,}597{,}484{,}987{\mbox{ copies of }}3\end{matrix}}}  </td><td>                                                                                                                                    3                                                                                                                    3                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        3                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  3                                                                                                                    3                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        3                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  3                                                                                                                    3                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        3                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                      7,625,597,484,987                                                       copies of                                                   3                                                          {\displaystyle {\begin{matrix}\underbrace {3_{}^{3^{{}^{.\,^{.\,^{.\,^{3}}}}}}} \\\underbrace {3_{}^{3^{{}^{.\,^{.\,^{.\,^{3}}}}}}} \\\underbrace {3_{}^{3^{{}^{.\,^{.\,^{.\,^{3}}}}}}} \\7{,}625{,}597{,}484{,}987{\mbox{ copies of }}3\end{matrix}}}  </td><td>                    3        ↑↑↑        b              {\displaystyle 3\uparrow \uparrow \uparrow b}  </td></tr><tr><th>4</th><td>3</td><td>                                                                                                                                    3                                                                                                                    3                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        3                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                      7,625,597,484,987                                                       copies of                                                   3                                                          {\displaystyle {\begin{matrix}\underbrace {3_{}^{3^{{}^{.\,^{.\,^{.\,^{3}}}}}}} \\7{,}625{,}597{,}484{,}987{\mbox{ copies of }}3\end{matrix}}}  </td><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              3                                                                                                                    .                                                                                                        .                                                                                            .                                                                                3                                                                    3                                        ⏟                                                                                                                                                                  3                                                                                                                    3                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        3                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                      7,625,597,484,987                                                       copies of                                                   3                                                          {\displaystyle {\begin{matrix}\underbrace {^{^{^{^{^{3}.}.}.}3}3} \\\underbrace {3_{}^{3^{{}^{.\,^{.\,^{.\,^{3}}}}}}} \\7{,}625{,}597{,}484{,}987{\mbox{ copies of }}3\end{matrix}}}  </td><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              3                                                                                                                    .                                                                                                        .                                                                                            .                                                                                3                                                                    3                                        ⏟                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            3                                                                                                                    .                                                                                                        .                                                                                            .                                                                                3                                                                    3                                        ⏟                                                                                                                                                                  3                                                                                                                    3                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        3                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                      7,625,597,484,987                                                       copies of                                                   3                                                          {\displaystyle {\begin{matrix}\underbrace {^{^{^{^{^{3}.}.}.}3}3} \\\underbrace {^{^{^{^{^{3}.}.}.}3}3} \\\underbrace {3_{}^{3^{{}^{.\,^{.\,^{.\,^{3}}}}}}} \\7{,}625{,}597{,}484{,}987{\mbox{ copies of }}3\end{matrix}}}  </td><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              3                                                                                                                    .                                                                                                        .                                                                                            .                                                                                3                                                                    3                                        ⏟                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            3                                                                                                                    .                                                                                                        .                                                                                            .                                                                                3                                                                    3                                        ⏟                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            3                                                                                                                    .                                                                                                        .                                                                                            .                                                                                3                                                                    3                                        ⏟                                                                                                                                                                  3                                                                                                                    3                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        3                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                      7,625,597,484,987                                                       copies of                                                   3                                                          {\displaystyle {\begin{matrix}\underbrace {^{^{^{^{^{3}.}.}.}3}3} \\\underbrace {^{^{^{^{^{3}.}.}.}3}3} \\\underbrace {^{^{^{^{^{3}.}.}.}3}3} \\\underbrace {3_{}^{3^{{}^{.\,^{.\,^{.\,^{3}}}}}}} \\7{,}625{,}597{,}484{,}987{\mbox{ copies of }}3\end{matrix}}}  </td><td>                    3        ↑↑↑↑        b              {\displaystyle 3\uparrow \uparrow \uparrow \uparrow b}  </td></tr></tbody></table>
<h3>Computing 4↑<i>n</i> <i>b</i></h3>
<p>We place the numbers 
  
    
      
        
          4
          
            b
          
        
      
    
    {\displaystyle 4^{b}}
  
 in the top row, and fill the left column with values 4. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.
</p>
Values of                     4                  ↑                      n                          b              {\displaystyle 4\uparrow ^{n}b}   = <a href="/facts/Hyperoperation/XNn1hneN">                              H                      n            +            2                          (        4        ,        b        )              {\displaystyle H_{n+2}(4,b)}  </a> = <a href="/facts/Hyperoperation/XNn1hneN">                    4        [        n        +        2        ]        b              {\displaystyle 4[n+2]b}  </a> = <a href="/facts/Conway_chained_arrow_notation/VHlJgdd8">4 → b → n</a><table><tbody><tr><th><i>b</i><i>ⁿ</i></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>formula</th></tr><tr><th>1</th><td>4</td><td>16</td><td>64</td><td>256</td><td>1024</td><td>                              4                      b                                {\displaystyle 4^{b}}  </td></tr><tr><th>2</th><td>4</td><td>256</td><td>                              4                      256                          ≈        1.34        ×                  10                      154                                {\displaystyle 4^{256}\approx 1.34\times 10^{154}}  </td><td>                              4                                    4                              256                                                    ≈                  10                      8.0            ×                          10                              153                                                          {\displaystyle 4^{4^{256}}\approx 10^{8.0\times 10^{153}}}  </td><td>                              4                                    4                                                4                                      256                                                                                            {\displaystyle 4^{4^{4^{256}}}}  </td><td>                    4        ↑↑        b              {\displaystyle 4\uparrow \uparrow b}  </td></tr><tr><th>3</th><td>4</td><td>                              4                                    4                              256                                                    ≈                  10                      8.0            ×                          10                              153                                                          {\displaystyle 4^{4^{256}}\approx 10^{8.0\times 10^{153}}}  </td><td>                                                                                                                                    4                                                                                                                    4                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        4                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                        4                                                            4                                              256                                                                                                                                   copies of                                                   4                                                          {\displaystyle {\begin{matrix}\underbrace {4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}} \\4^{4^{256}}{\mbox{ copies of }}4\end{matrix}}}  </td><td>                                                                                                                                    4                                                                                                                    4                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        4                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  4                                                                                                                    4                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        4                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                        4                                                            4                                              256                                                                                                                                   copies of                                                   4                                                          {\displaystyle {\begin{matrix}\underbrace {4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}} \\\underbrace {4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}} \\4^{4^{256}}{\mbox{ copies of }}4\end{matrix}}}  </td><td>                                                                                                                                    4                                                                                                                    4                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        4                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  4                                                                                                                    4                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        4                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  4                                                                                                                    4                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        4                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                        4                                                            4                                              256                                                                                                                                   copies of                                                   4                                                          {\displaystyle {\begin{matrix}\underbrace {4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}} \\\underbrace {4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}} \\\underbrace {4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}} \\4^{4^{256}}{\mbox{ copies of }}4\end{matrix}}}  </td><td>                    4        ↑↑↑        b              {\displaystyle 4\uparrow \uparrow \uparrow b}  </td></tr><tr><th>4</th><td>4</td><td>                                                                                                                                    4                                                                                                                    4                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        4                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  4                                                                                                                    4                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        4                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                        4                                                            4                                              256                                                                                                                                   copies of                                                   4                                                          {\displaystyle {\begin{matrix}\underbrace {4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}} \\\underbrace {4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}} \\4^{4^{256}}{\mbox{ copies of }}4\end{matrix}}}  </td><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              4                                                                                                                    .                                                                                                        .                                                                                            .                                                                                4                                                                    4                                        ⏟                                                                                                                                                                  4                                                                                                                    4                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        4                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  4                                                                                                                    4                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        4                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                        4                                                            4                                              256                                                                                                                                   copies of                                                   4                                                          {\displaystyle {\begin{matrix}\underbrace {^{^{^{^{^{4}.}.}.}4}4} \\\underbrace {4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}} \\\underbrace {4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}} \\4^{4^{256}}{\mbox{ copies of }}4\end{matrix}}}  </td><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              4                                                                                                                    .                                                                                                        .                                                                                            .                                                                                4                                                                    4                                        ⏟                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            4                                                                                                                    .                                                                                                        .                                                                                            .                                                                                4                                                                    4                                        ⏟                                                                                                                                                                  4                                                                                                                    4                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        4                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  4                                                                                                                    4                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        4                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                        4                                                            4                                              256                                                                                                                                   copies of                                                   4                                                          {\displaystyle {\begin{matrix}\underbrace {^{^{^{^{^{4}.}.}.}4}4} \\\underbrace {^{^{^{^{^{4}.}.}.}4}4} \\\underbrace {4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}} \\\underbrace {4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}} \\4^{4^{256}}{\mbox{ copies of }}4\end{matrix}}}  </td><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              4                                                                                                                    .                                                                                                        .                                                                                            .                                                                                4                                                                    4                                        ⏟                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            4                                                                                                                    .                                                                                                        .                                                                                            .                                                                                4                                                                    4                                        ⏟                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            4                                                                                                                    .                                                                                                        .                                                                                            .                                                                                4                                                                    4                                        ⏟                                                                                                                                                                  4                                                                                                                    4                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        4                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  4                                                                                                                    4                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        4                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                        4                                                            4                                              256                                                                                                                                   copies of                                                   4                                                          {\displaystyle {\begin{matrix}\underbrace {^{^{^{^{^{4}.}.}.}4}4} \\\underbrace {^{^{^{^{^{4}.}.}.}4}4} \\\underbrace {^{^{^{^{^{4}.}.}.}4}4} \\\underbrace {4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}} \\\underbrace {4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}} \\4^{4^{256}}{\mbox{ copies of }}4\end{matrix}}}  </td><td>                    4        ↑↑↑↑        b              {\displaystyle 4\uparrow \uparrow \uparrow \uparrow b}  </td></tr></tbody></table>
<h3>Computing 10↑<i>n</i> <i>b</i></h3>
<p>We place the numbers 
  
    
      
        
          10
          
            b
          
        
      
    
    {\displaystyle 10^{b}}
  
 in the top row, and fill the left column with values 10. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.
</p>
Values of                     10                  ↑                      n                          b              {\displaystyle 10\uparrow ^{n}b}   = <a href="/facts/Hyperoperation/XNn1hneN">                              H                      n            +            2                          (        10        ,        b        )              {\displaystyle H_{n+2}(10,b)}  </a> = <a href="/facts/Hyperoperation/XNn1hneN">                    10        [        n        +        2        ]        b              {\displaystyle 10[n+2]b}  </a> = <a href="/facts/Conway_chained_arrow_notation/VHlJgdd8">10 → b → n</a><table><tbody><tr><th><i>b</i><i>ⁿ</i></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>formula</th></tr><tr><th>1</th><td>10</td><td>100</td><td>1,000</td><td>10,000</td><td>100,000</td><td>                              10                      b                                {\displaystyle 10^{b}}  </td></tr><tr><th>2</th><td>10</td><td>10,000,000,000</td><td>                              10                      10            ,            000            ,            000            ,            000                                {\displaystyle 10^{10,000,000,000}}  </td><td>                              10                                    10                              10                ,                000                ,                000                ,                000                                                          {\displaystyle 10^{10^{10,000,000,000}}}  </td><td>                              10                                    10                                                10                                      10                    ,                    000                    ,                    000                    ,                    000                                                                                            {\displaystyle 10^{10^{10^{10,000,000,000}}}}  </td><td>                    10        ↑↑        b              {\displaystyle 10\uparrow \uparrow b}  </td></tr><tr><th>3</th><td>10</td><td>                                                                                                                                    10                                                                                                                    10                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        10                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                      10                                                       copies of                                                   10                                                          {\displaystyle {\begin{matrix}\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\10{\mbox{ copies of }}10\end{matrix}}}  </td><td>                                                                                                                                    10                                                                                                                    10                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        10                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  10                                                                                                                    10                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        10                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                      10                                                       copies of                                                   10                                                          {\displaystyle {\begin{matrix}\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\10{\mbox{ copies of }}10\end{matrix}}}  </td><td>                                                                                                                                    10                                                                                                                    10                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        10                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  10                                                                                                                    10                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        10                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  10                                                                                                                    10                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        10                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                      10                                                       copies of                                                   10                                                          {\displaystyle {\begin{matrix}\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\10{\mbox{ copies of }}10\end{matrix}}}  </td><td>                                                                                                                                    10                                                                                                                    10                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        10                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  10                                                                                                                    10                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        10                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  10                                                                                                                    10                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        10                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                                                                                  10                                                                                                                    10                                                                                                                                                                                .                                                                                                                                        .                                                                                                                                                        .                                                                                                                                                                        10                                                                                                                                                                                                                                                                                                                                                                                                        ⏟                                                                                                      10                                                       copies of                                                   10                                                          {\displaystyle {\begin{matrix}\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\10{\mbox{ copies of }}10\end{matrix}}}  </td><td>                    10        ↑↑↑        b              {\displaystyle 10\uparrow \uparrow \uparrow b}  </td></tr><tr><th>4</th><td>10</td><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              10                                                                                                                    .                                                                                                        .                                                                                            .                                                                                10                                                                    10                                        ⏟                                                                                                      10                                                       copies of                                                   10                                                          {\displaystyle {\begin{matrix}\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\10{\mbox{ copies of }}10\end{matrix}}}  </td><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              10                                                                                                                    .                                                                                                        .                                                                                            .                                                                                10                                                                    10                                        ⏟                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            10                                                                                                                    .                                                                                                        .                                                                                            .                                                                                10                                                                    10                                        ⏟                                                                                                      10                                                       copies of                                                   10                                                          {\displaystyle {\begin{matrix}\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\10{\mbox{ copies of }}10\end{matrix}}}  </td><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              10                                                                                                                    .                                                                                                        .                                                                                            .                                                                                10                                                                    10                                        ⏟                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            10                                                                                                                    .                                                                                                        .                                                                                            .                                                                                10                                                                    10                                        ⏟                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            10                                                                                                                    .                                                                                                        .                                                                                            .                                                                                10                                                                    10                                        ⏟                                                                                                      10                                                       copies of                                                   10                                                          {\displaystyle {\begin{matrix}\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\10{\mbox{ copies of }}10\end{matrix}}}  </td><td>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              10                                                                                                                    .                                                                                                        .                                                                                            .                                                                                10                                                                    10                                        ⏟                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            10                                                                                                                    .                                                                                                        .                                                                                            .                                                                                10                                                                    10                                        ⏟                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            10                                                                                                                    .                                                                                                        .                                                                                            .                                                                                10                                                                    10                                        ⏟                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            10                                                                                                                    .                                                                                                        .                                                                                            .                                                                                10                                                                    10                                        ⏟                                                                                                      10                                                       copies of                                                   10                                                          {\displaystyle {\begin{matrix}\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\10{\mbox{ copies of }}10\end{matrix}}}  </td><td>                    10        ↑↑↑↑        b              {\displaystyle 10\uparrow \uparrow \uparrow \uparrow b}  </td></tr></tbody></table>
<p>For 2 ≤ <i>b</i> ≤ 9 the numerical order of the numbers 
  
    
      
        10
        
          ↑
          
            n
          
        
        b
      
    
    {\displaystyle 10\uparrow ^{n}b}
  
 is the <a href="/facts/Lexicographical_order/xQANy8Br">lexicographical order</a> with <i>n</i> as the most significant number, so for the numbers of these 8 columns the numerical order is simply line-by-line. The same applies for the numbers in the 97 columns with 3 ≤ <i>b</i> ≤ 99, and if we start from <i>n</i> = 1 even for 3 ≤ <i>b</i> ≤ 9,999,999,999.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Primitive_recursion/4z97gRnl">Primitive recursion</a></li>
<li><a href="/facts/Hyperoperation/XNn1hneN">Hyperoperation</a></li>
<li><a href="/facts/Busy_beaver/qfdJPkVu">Busy beaver</a></li>
<li><a href="/facts/Cutler%27s_bar_notation/s5kIxMR3">Cutler's bar notation</a></li>
<li><a href="/facts/Tetration/5rKeVVYv">Tetration</a></li>
<li><a href="/facts/Pentation/YKhnIWPG">Pentation</a></li>
<li><a href="/facts/Ackermann_function/WQNpCWHn">Ackermann function</a></li>
<li><a href="/facts/Graham%27s_number/7IJYxN2x">Graham's number</a></li>
<li><a href="/facts/Steinhaus%E2%80%93Moser_notation/Wf1zyukm">Steinhaus–Moser notation</a></li></ul>
<h2 id="notes">Notes</h2>

<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/KnuthUp-ArrowNotation.html">"Knuth Up-Arrow Notation"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li>
<li>Robert Munafo, <i><a href="http://www.mrob.com/pub/math/largenum-3.html#hyper5">Large Numbers: Higher hyper operators</a></i></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Knuth, Donald E. (1976). "Mathematics and Computer Science: Coping with Finiteness". Science. 194 (4271): 1235–1242. Bibcode:1976Sci...194.1235K. doi:10.1126/science.194.4271.1235. PMID 17797067. S2CID 1690489. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>R. L. Goodstein (Dec 1947). "Transfinite Ordinals in Recursive Number Theory". Journal of Symbolic Logic. 12 (4): 123–129. doi:10.2307/2266486. JSTOR 2266486. S2CID 1318943. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>For more details, see Powers of zero. <a href="/wiki/Exponentiation#Powers_of_zero" target="_blank">/wiki/Exponentiation#Powers_of_zero</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Keep in mind that Knuth did not define the operator 
  
    
      
        
          ↑
          
            0
          
        
      
    
    {\displaystyle \uparrow ^{0}}
  
. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>For more details, see Powers of zero. <a href="/wiki/Exponentiation#Powers_of_zero" target="_blank">/wiki/Exponentiation#Powers_of_zero</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>For more details, see Zero to the power of zero. <a href="/wiki/Zero_to_the_power_of_zero" target="_blank">/wiki/Zero_to_the_power_of_zero</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>For more details, see Powers of zero. <a href="/wiki/Exponentiation#Powers_of_zero" target="_blank">/wiki/Exponentiation#Powers_of_zero</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>For more details, see Zero to the power of zero. <a href="/wiki/Zero_to_the_power_of_zero" target="_blank">/wiki/Zero_to_the_power_of_zero</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
</ol>

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

Knuth's up-arrow notation