Knuth's up-arrow notation

In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976.^[1]

In his 1947 paper,^[2] R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations. Goodstein also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation. The sequence starts with a unary operation (the successor function with n = 0), and continues with the binary operations of addition (n = 1), multiplication (n = 2), exponentiation (n = 3), tetration (n = 4), pentation (n = 5), etc. Various notations have been used to represent hyperoperations. One such notation is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_n(a,b)} . Knuth's up-arrow notation $\uparrow$ is another. For example:

the single arrow $\uparrow$ represents exponentiation (iterated multiplication) $2\uparrow 4=H_{3}(2,4)=2\times (2\times (2\times 2))=2^{4}=16$
the double arrow $\uparrow \uparrow$ represents tetration (iterated exponentiation) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \uparrow\uparrow 4 = H_4(2,4) = 2 \uparrow (2 \uparrow (2 \uparrow 2))= 2^{2^{2^{2}}} = 2^{16} = 65,536}
the triple arrow Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \uparrow\uparrow\uparrow} represents pentation (iterated tetration) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 2 \uparrow\uparrow\uparrow 4 = H_5(2,4) = 2 \uparrow\uparrow (2 \uparrow\uparrow (2 \uparrow\uparrow 2 ))\\ &= 2 \uparrow\uparrow (2 \uparrow\uparrow (2 \uparrow 2 ))\\ &= 2 \uparrow\uparrow (2 \uparrow\uparrow 4 )\\ &= \underbrace{2 \uparrow (2 \uparrow (2 \uparrow\dots ))} \; = \; \underbrace{ \; 2^{2^{\cdots^2}}}\\ & \;\;\;\;\; 2 \uparrow\uparrow 4 \mbox{ copies of } 2 \;\;\;\;\; \mbox{65,536 2s}\\ \end{align}}

The general definition of the up-arrow notation is as follows (for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ge 0, n \ge 1, b \ge 0} ): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\uparrow^nb = H_{n+2}(a,b) = a[n+2]b.} Here, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \uparrow^n} stands for n arrows, so for example Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \uparrow\uparrow\uparrow\uparrow 3 = 2\uparrow^4 3.} The square brackets are another notation for hyperoperations.

Introduction

The hyperoperations naturally extend the arithmetic operations of addition and multiplication as follows. Addition by a natural number is defined as iterated incrementation:

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

Multiplication by a natural number is defined as iterated addition:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} H_2(a,b) = a\times b = & \underbrace{a+a+\dots+a} \\ & b\mbox{ copies of }a \end{matrix} }

For example,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} 4\times 3 & = & \underbrace{4+4+4} & = & 12\\ & & 3\mbox{ copies of }4 \end{matrix} }

Exponentiation for a natural power Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} is defined as iterated multiplication, which Knuth denoted by a single up-arrow:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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} }

For example,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} 4\uparrow 3= 4^3 = & \underbrace{4\times 4\times 4} & = & 64\\ & 3\mbox{ copies of }4 \end{matrix} }

Tetration is defined as iterated exponentiation, which Knuth denoted by a “double arrow”:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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} }

For example,

{\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}}

Expressions are evaluated from right to left, as the operators are defined to be right-associative.

According to this definition,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\uparrow\uparrow 2=3^3=27 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\uparrow\uparrow 3=3^{3^3}=3^{27}=7,625,597,484,987 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\uparrow\uparrow 4=3^{3^{3^3}}=3^{3^{27}}=3^{7625597484987}\approx 1.2580143\times 10^{3638334640024} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.

This already leads to some fairly large numbers, but the hyperoperator sequence does not stop here.

Pentation, defined as iterated tetration, is represented by the “triple arrow”:

{\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}}

Hexation, defined as iterated pentation, is represented by the “quadruple arrow”:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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} }

and so on. The general rule is that an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -arrow operator expands into a right-associative series of (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n - 1} )-arrow operators. Symbolically,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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} }

Examples:

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} 3\uparrow\uparrow\uparrow3 = 3\uparrow\uparrow(3\uparrow\uparrow3) = 3\uparrow\uparrow(3\uparrow3\uparrow3) = & \underbrace{3_{}\uparrow 3\uparrow\dots\uparrow 3} \\ & 3\uparrow3\uparrow3\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} }

Notation

In expressions such as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^b} , the notation for exponentiation is usually to write the exponent Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} as a superscript to the base number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} . But many environments — such as programming languages and plain-text e-mail — do not support superscript typesetting. People have adopted the linear notation $a\uparrow b$ for such environments; the up-arrow suggests 'raising to the power of'. If the character set does not contain an up arrow, the caret (^) is used instead.

The superscript notation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^b} doesn't lend itself well to generalization, which explains why Knuth chose to work from the inline notation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \uparrow b} instead.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \uparrow^n b} is a shorter alternative notation for n uparrows. Thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \uparrow^4 b = a \uparrow \uparrow \uparrow \uparrow b} .

Writing out up-arrow notation in terms of powers

Attempting to write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \uparrow \uparrow b} using the familiar superscript notation gives a power tower.

For example: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \uparrow \uparrow 4 = a \uparrow (a \uparrow (a \uparrow a)) = a^{a^{a^a}}}

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.

a\uparrow \uparrow b=\underbrace {a^{a^{.^{.^{.{a}}}}}} _{b}

Continuing with this notation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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} }}}

Again, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} is a variable or is too large, the stack might be written using dots and a note indicating its height.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \uparrow \uparrow \uparrow b = \left. \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\} b}

Furthermore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}

And more generally:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}}

This might be carried out indefinitely to represent Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \uparrow^n b} as iterated exponentiation of iterated exponentiation for any $a$ , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} (although it clearly becomes rather cumbersome).

Using tetration

The Rudy Rucker notation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{b}a} for tetration allows us to make these diagrams slightly simpler while still employing a geometric representation (we could call these tetration towers).

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \uparrow \uparrow b = { }^{b}a}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \uparrow \uparrow \uparrow b = \underbrace{^{^{^{^{^{a}.}.}.}a}a}_{b}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \uparrow \uparrow \uparrow \uparrow b = \left. \underbrace{^{^{^{^{^{a}.}.}.}a}a}_{ \underbrace{^{^{^{^{^{a}.}.}.}a}a}_{ \underbrace{\vdots}_{a} }} \right\} b}

Finally, as an example, the fourth Ackermann number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 \uparrow^4 4} could be represented as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 }}}

Generalizations

Some numbers are so large that multiple arrows of Knuth's up-arrow notation become too cumbersome; then an n-arrow operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \uparrow^n} is useful (and also for descriptions with a variable number of arrows), or equivalently, hyper operators.

Some numbers are so large that even that notation is not sufficient. The Conway chained arrow notation 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.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} a\uparrow^n b & = & a [n+2] b & = & a\to b\to n \\ \mbox{(Knuth)} & & \mbox{(hyperoperation)} & & \mbox{(Conway)} \end{matrix} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6\uparrow\uparrow4} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \underbrace{6^{6^{.^{.^{.^{6}}}}}}_{4}} , Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6\uparrow\uparrow4} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6^{6^{6^{6}}}} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6^{6^{46,656}}} , Thus the result comes out with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \underbrace{6^{6^{.^{.^{.^{6}}}}}}_{4}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10\uparrow(3\times10\uparrow(3\times10\uparrow15)+3)} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \underbrace{100000...000}_{ \underbrace{300000...003}_{\underbrace{300000...000}_{15} }}} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{3\times10^{3\times10^{15}}+3}} (Petillion)

Even faster-growing functions can be categorized using an ordinal analysis called the fast-growing hierarchy. The fast-growing hierarchy uses successive function iteration and diagonalization to systematically create faster-growing functions from some base function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} . For the standard fast-growing hierarchy using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_0(x) = x+1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_2(x)} already exhibits exponential growth, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_3(x)} is comparable to tetrational growth and is upper-bounded by a function involving the first four hyperoperators;. Then, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_\omega(x)} is comparable to the Ackermann function, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{\omega + 1}(x)} is already beyond the reach of indexed arrows but can be used to approximate Graham's number, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{\omega^2}(x)} is comparable to arbitrarily-long Conway chained arrow notation.

These functions are all computable. Even faster computable functions, such as the Goodstein sequence and the TREE sequence 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 Busy Beaver, whose very nature will be completely out of reach from any up-arrow, or even any ordinal-based analysis.

Definition

Without reference to hyperoperation the up-arrow operators can be formally defined by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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} }

for all integers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b, n} with $a\geq 0,n\geq 1,b\geq 0$ ^{[nb 1]}.

This definition uses exponentiation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a\uparrow^1 b = a\uparrow b = a^b)} as the base case, and tetration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a\uparrow^2 b = a\uparrow\uparrow b)} as repeated exponentiation. This is equivalent to the hyperoperation sequence except it omits the three more basic operations of succession, addition and multiplication.

One can alternatively choose multiplication Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a\uparrow^0 b = a \times b)} as the base case and iterate from there. Then exponentiation becomes repeated multiplication. The formal definition would be

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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} }

for all integers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b, n} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ge 0, n \ge 0, b \ge 0} .

Note, however, that Knuth did not define the "nil-arrow" (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_n(a, b) = a [n] b = a \uparrow^{n-2}b\text{ for } n \ge 0.}

The up-arrow operation is a right-associative operation, that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \uparrow b \uparrow c} is understood to be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \uparrow (b \uparrow c)} , instead of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a \uparrow b) \uparrow c} . If ambiguity is not an issue parentheses are sometimes dropped.

Tables of values

Computing 0↑ⁿ b

Computing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\uparrow^n b = H_{n+2}(0,b) = 0[n+2]b} results in

0, when n = 0 ^{[nb 2]}

1, when n = 1 and b = 0 ^{[nb 1]}^{[nb 3]}

0, when n = 1 and b > 0 ^{[nb 1]}^{[nb 3]}

1, when n > 1 and b is even (including 0)

0, when n > 1 and b is odd

Computing 2↑ⁿ b

Computing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\uparrow^n b} can be restated in terms of an infinite table. We place the numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.

Values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\uparrow^n b} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{n+2}(2,b)} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2[n+2]b} = 2 → b → n
b ⁿ	1	2	3	4	5	6	formula
1	2	4	8	16	32	64	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^b}
2	2	4	16	65536	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^{65{,}536}\approx 2.0 \times 10^{19{,}728}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^{2^{65{,}536}}\approx 10^{6.0 \times 10^{19{,}727}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\uparrow\uparrow b}
3	2	4	65536	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{2_{}^{2^{{}^{.\,^{.\,^{.\,^2}}}}}} \\ 65{,}536\mbox{ copies of }2 \end{matrix} }	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}}\\ \underbrace{2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}}\\ 65{,}536\mbox{ copies of }2 \end{matrix}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}}\\ \underbrace{2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}}\\ \underbrace{2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}}\\ 65{,}536\mbox{ copies of }2 \end{matrix}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\uparrow\uparrow\uparrow b}
4	2	4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{2_{}^{2^{{}^{.\,^{.\,^{.\,^2}}}}}}\\ 65{,}536\mbox{ copies of }2 \end{matrix}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{^{^{^{^{^{2}.}.}.}2}2}\\ \underbrace{2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}}\\ 65{,}536\mbox{ copies of }2 \end{matrix}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{^{^{^{^{^{2}.}.}.}2}2}\\ \underbrace{^{^{^{^{^{2}.}.}.}2}2}\\ \underbrace{2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}}\\ 65{,}536\mbox{ copies of }2 \end{matrix}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\uparrow\uparrow\uparrow\uparrow b}

The table is the same as that of the Ackermann function, except for a shift in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} , and an addition of 3 to all values.

Computing 3↑ⁿ b

We place the numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.

Values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\uparrow^n b} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{n+2}(3,b)} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3[n+2]b} = 3 → b → n
b ⁿ	1	2	3	4	5	formula
1	3	9	27	81	243	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3^b}
2	3	27	7,625,597,484,987	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3^{7{,}625{,}597{,}484{,}987}\approx 1.3 \times 10^{3{,}638{,}334{,}640{,}024}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3^{3^{7{,}625{,}597{,}484{,}987}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\uparrow\uparrow b}
3	3	7,625,597,484,987	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{3_{}^{3^{{}^{.\,^{.\,^{.\,^3}}}}}}\\ 7{,}625{,}597{,}484{,}987\mbox{ copies of }3 \end{matrix}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{3_{}^{3^{{}^{.\,^{.\,^{.\,^{3}}}}}}}\\ \underbrace{3_{}^{3^{{}^{.\,^{.\,^{.\,^{3}}}}}}}\\ 7{,}625{,}597{,}484{,}987\mbox{ copies of }3 \end{matrix}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\uparrow\uparrow\uparrow b}
4	3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{3_{}^{3^{{}^{.\,^{.\,^{.\,^3}}}}}}\\ 7{,}625{,}597{,}484{,}987\mbox{ copies of }3 \end{matrix}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{^{^{^{^{^{3}.}.}.}3}3}\\ \underbrace{3_{}^{3^{{}^{.\,^{.\,^{.\,^3}}}}}}\\ 7{,}625{,}597{,}484{,}987\mbox{ copies of }3 \end{matrix}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\uparrow\uparrow\uparrow\uparrow b}

Computing 4↑ⁿ b

We place the numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.

Values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4\uparrow^n b} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{n+2}(4,b)} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4[n+2]b} = 4 → b → n
b ⁿ	1	2	3	4	5	formula
1	4	16	64	256	1024	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4^b}
2	4	256	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4^{256}\approx 1.34 \times 10^{154}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4^{4^{256}}\approx 10^{8.0 \times 10^{153}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4^{4^{4^{256}}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4\uparrow\uparrow b}
3	4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4^{4^{256}}\approx 10^{8.0 \times 10^{153}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{4_{}^{4^{{}^{.\,^{.\,^{.\,^4}}}}}}\\ 4^{4^{256}}\mbox{ copies of }4 \end{matrix}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{4_{}^{4^{{}^{.\,^{.\,^{.\,^4}}}}}}\\ \underbrace{4_{}^{4^{{}^{.\,^{.\,^{.\,^4}}}}}}\\ 4^{4^{256}}\mbox{ copies of }4 \end{matrix}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{4_{}^{4^{{}^{.\,^{.\,^{.\,^4}}}}}}\\ \underbrace{4_{}^{4^{{}^{.\,^{.\,^{.\,^4}}}}}}\\ \underbrace{4_{}^{4^{{}^{.\,^{.\,^{.\,^4}}}}}}\\ 4^{4^{256}}\mbox{ copies of }4 \end{matrix}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4\uparrow\uparrow\uparrow b}
4	4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}}\\ \underbrace{4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}}\\ 4^{4^{256}}\mbox{ copies of }4 \end{matrix}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{^{^{^{^{^{4}.}.}.}4}4}\\ \underbrace{4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}}\\ \underbrace{4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}}\\ 4^{4^{256}}\mbox{ copies of }4 \end{matrix}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4\uparrow\uparrow\uparrow\uparrow b}

Computing 10↑ⁿ b

We place the numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.

Values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10\uparrow^n b} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{n+2}(10,b)} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10[n+2]b} = 10 → b → n
b ⁿ	1	2	3	4	5	formula
1	10	100	1,000	10,000	100,000	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^b}
2	10	10,000,000,000	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{10,000,000,000}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{10^{10,000,000,000}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{10^{10^{10,000,000,000}}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10\uparrow\uparrow b}
3	10	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\ 10\mbox{ copies of }10 \end{matrix}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\ \underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\ 10\mbox{ copies of }10 \end{matrix}}	${\begin{matrix}\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\10{\mbox{ copies of }}10\end{matrix}}$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10\uparrow\uparrow\uparrow b}
4	10	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{^{^{^{^{^{10}.}.}.}10}10}\\ 10\mbox{ copies of }10 \end{matrix}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{^{^{^{^{^{10}.}.}.}10}10}\\ \underbrace{^{^{^{^{^{10}.}.}.}10}10}\\ 10\mbox{ copies of }10 \end{matrix}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \underbrace{^{^{^{^{^{10}.}.}.}10}10}\\ \underbrace{^{^{^{^{^{10}.}.}.}10}10}\\ \underbrace{^{^{^{^{^{10}.}.}.}10}10}\\ 10\mbox{ copies of }10 \end{matrix}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10\uparrow\uparrow\uparrow\uparrow b}

For 2 ≤ b ≤ 9 the numerical order of the numbers $10\uparrow ^{n}b$ is the lexicographical order with n 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 ≤ b ≤ 99, and if we start from n = 1 even for 3 ≤ b ≤ 9,999,999,999.

Notes

^ ^1.0 ^1.1 ^1.2 For more details, see Powers of zero.
^ Keep in mind that Knuth did not define the operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \uparrow^0} .
^ ^3.0 ^3.1 For more details, see Zero to the power of zero.

References

^ 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.
^ 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.

External links

Weisstein, Eric W. "Knuth Up-Arrow Notation". MathWorld.
Robert Munafo, Large Numbers: Higher hyper operators

[corona2-3] 1.0 ^1.1 ^1.2 For more details, see Powers of zero.

[corona1-4] Keep in mind that Knuth did not define the operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \uparrow^0} .

[corona3-5] 3.0 ^3.1 For more details, see Zero to the power of zero.

[1] 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.

[2] 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.

[1]

[2]

[nb 1]

[nb 2]

[nb 3]

v t e Hyperoperations
Primary	Successor (0) Addition (1) Multiplication (2) Exponentiation (3) Tetration (4) Pentation (5) Hexation (6)
Inverse for left argument	Predecessor (0) Subtraction (1) Division (2) Root extraction (3) Super-root (4)
Inverse for right argument	Predecessor (0) Subtraction (1) Division (2) Logarithm (3) Super-logarithm (4)
Related articles	Ackermann function Conway chained arrow notation Grzegorczyk hierarchy Knuth's up-arrow notation Steinhaus–Moser notation

v t e Donald Knuth
Publications	The Art of Computer Programming "The Complexity of Songs" Computers and Typesetting Concrete Mathematics Surreal Numbers Things a Computer Scientist Rarely Talks About Selected papers series
Software	TeX Metafont MIXAL (MIX MMIX)
Fonts	AMS Euler Computer Modern Concrete Roman
Literate programming	WEB CWEB
Algorithms	Knuth's Algorithm X Knuth–Bendix completion algorithm Knuth–Morris–Pratt algorithm Knuth shuffle Robinson–Schensted–Knuth correspondence Trabb Pardo–Knuth algorithm Generalization of Dijkstra's algorithm Knuth's Simpath algorithm
Other	Dancing Links Knuth reward check Knuth Prize Knuth's up-arrow notation Man or boy test Quater-imaginary base -yllion Potrzebie system of weights and measures

Introduction

Notation

Writing out up-arrow notation in terms of powers

Using tetration

Generalizations

Definition

Tables of values

Computing 0↑n b

Computing 2↑n b

Computing 3↑n b

Computing 4↑n b

Computing 10↑n b

See also

Notes

References

External links

Computing 0↑ⁿ b

Computing 2↑ⁿ b

Computing 3↑ⁿ b

Computing 4↑ⁿ b

Computing 10↑ⁿ b