Inmathematics, thehyperoperation sequence is an infinitesequence of arithmetic operations (calledhyperoperations in this context)^[1][2][3] that starts with aunary operation (thesuccessor function withn = 0). The sequence continues with thebinary operations ofaddition (n = 1),multiplication (n = 2), andexponentiation (n = 3).^{[nb 1]} After that, the sequence proceeds with further binary operations extending beyond exponentiation, usingright-associativity. For the operations beyond exponentiation, thenth member of this sequence is named byReuben Goodstein after theGreek prefix ofn suffixed with-ation (such astetration (n = 4), pentation (n = 5), hexation (n = 6), etc.)^[7] and can be written usingn − 2 arrows inKnuth's up-arrow notation.Each hyperoperation may be understoodrecursively in terms of the previous one by:

${\displaystyle a[n]b=\underset{{\displaystyle b\text{ copies of }a}}{\underbrace{a[n-1](a[n-1](a[n-1](\cdots a[n-1](a[n-1](a[n-1]a))\cdots )))}},n\ge 2}$

It may also be defined according to the recursion rule part of the definition, as in Knuth's up-arrow version of theAckermann function:

${\displaystyle a[n]b=a[n-1]\left(a[n]\left(b-1\right)\right),n\ge 1}$

This can be used to easily show numbers much larger than those whichscientific notation can, such asSkewes's number andgoogolplexplex (e.g.${\displaystyle 50[50]50}$ is much larger than Skewes's number and googolplexplex), but there are some numbers which even they cannot easily show, such asGraham's number andTREE(3).^[14]

This recursion rule is common to many variants of hyperoperations.

Thehyperoperation sequence is thesequence ofbinary operations${\displaystyle H_n:({\mathbb{N}}_0{)}^2\to {\mathbb{N}}_0}$ definedrecursively as follows:Forn = 0, 1, 2, 3, this definition reproduces the basic arithmetic operations ofsuccessor (which is a unary operation),addition,multiplication, andexponentiation, respectively, asfor all nonnegative integersa andb. The hyperoperations can thus be seen as an answer to the question "what's next?" in thesequence of functions that begins successor, addition, multiplication, exponentiation.Just as integer multiplication is defined as iterated addition and integer exponentiation is defined by iterated multiplication, the next hyperoperation,tetration, is defined by iterated exponentiation; for example,${\displaystyle H_4(a,3)=\mathrm{tetration}(a,3)=a^{a^a}}$ is a power tower of threeas, and${\displaystyle H_4(a,4)=\mathrm{tetration}(a,4)=a^{a^{a^a}}}$. Likewise the fifth hyperoperationpentation, is defined by iterated tetration, so that${\displaystyle H_5(a,3)=\mathrm{tetration}(a,\mathrm{tetration}(a,a))}$.

The parameters of the hyperoperation hierarchy are sometimes referred to by their analogous exponentiation term;^[15] soa is thebase,b is theexponent (orhyperexponent),^[13] andn is therank (orgrade).^[8] In general,${\displaystyle H_n(a,b)}$ may be read as "thebthn-ation ofa", so that${\displaystyle H_4(7,9)}$ is read as "the 9th tetration of 7", and${\displaystyle H_{123}(456,789)}$ is read as "the 789th 123-ation of 456".

An alternative way of writing hyperoperations is the compact notation${\displaystyle a[n]b}$ for${\displaystyle H_n(a,b)}$. In this notation, exponentiation is denoted${\displaystyle a[3]b=a^b}$, tetration is denoted${\displaystyle a[4]b}$ (so that${\displaystyle a[4]3=a^{a^a}}$, pentation is denoted${\displaystyle a[5]b}$, and so on. Hyperoperations can also be expressed usingKnuth's up-arrow notation. In this notation,${\displaystyle a\uparrow b}$ represents the exponentiation function${\displaystyle a^b}$,${\displaystyle a\uparrow \uparrow b}$ represents tetration,${\displaystyle a\uparrow \uparrow \uparrow b}$ or${\displaystyle a{\uparrow }^{3}b}$ represents the pentation${\displaystyle a[5]b}$, and more generally${\displaystyle H_n(a,b)=a{\uparrow }^{n-2}b}$ for${\displaystyle n\ge 0.}$ Yet another alternative isConway chained arrow notation. In this notation, one has${\displaystyle H_n(a,b)=a[n]b=a\to b\to n-2}$, so that (for example)${\displaystyle a[5]b=a\to b\to 3}$.^[16]

Below is a list of the first seven (0th to 6th) hyperoperations (0⁰ is defined as 1).

n	Operation, Hn(a,b)	Definition	Names	Domain
0	${\displaystyle b+1}$ or${\displaystyle a[0]b}$	${\displaystyle \underset{{\displaystyle b\text{ copies of 1}}}{\underbrace{1+1+1+\cdots +1+1+1}}+1}$	Increment,successor, zeration, hyper0	Arbitrary
1	${\displaystyle a+b}$ or${\displaystyle a[1]b}$	${\displaystyle a+\underset{{\displaystyle b\text{ copies of 1}}}{\underbrace{1+1+1+\cdots +1+1+1}}}$	Addition, hyper1
2	${\displaystyle a\times b}$ or${\displaystyle a[2]b}$	${\displaystyle \underset{{\displaystyle b\text{ copies of }a}}{\underbrace{a+a+a+\cdots +a+a+a}}}$	Multiplication, hyper2
3	${\displaystyle a^b}$ or${\displaystyle a[3]b}$	${\displaystyle \underset{{\displaystyle b\text{ copies of }a}}{\underbrace{a\times a\times a\times \cdots \times a\times a\times a}}}$	Exponentiation, hyper3	b real, with some multivalued extensions tocomplex numbers
4	${\displaystyle {}^{b}a}$ or${\displaystyle a[4]b}$	${\displaystyle \underset{{\displaystyle b\text{ copies of }a}}{\underbrace{a[3](a[3](a[3](\cdots [3](a[3](a[3]a))\cdots )))}}}$	Tetration, hyper4	a ≥ 0 or an integer,b an integer ≥ −1[nb 2] (with some proposed extensions)
5	${\displaystyle {}_{b}a}$ or${\displaystyle a[5]b}$	${\displaystyle \underset{{\displaystyle b\text{ copies of }a}}{\underbrace{a[4](a[4](a[4](\cdots [4](a[4](a[4]a))\cdots )))}}}$	Pentation, hyper5	a,b integers ≥ −1[nb 2]
6	${\displaystyle a[6]b}$	${\displaystyle \underset{{\displaystyle b\text{ copies of }a}}{\underbrace{a[5](a[5](a[5](\cdots [5](a[5](a[5]a))\cdots )))}}}$	Hexation, hyper6

H_n(0,b) =

b + 1, whenn = 0

b, whenn = 1

0, whenn = 2

1, whenn = 3 andb = 0^{[nb 3]}

0, whenn = 3 andb > 0^{[nb 3]}

1, whenn > 3 andb is even (including 0)

0, whenn > 3 andb is odd

H_n(1,b) =

b, whenn = 2

1, whenn ≥ 3

H_n(a, 0) =

0, whenn = 2

1, whenn = 0, orn ≥ 3

a, whenn = 1

H_n(a, 1) =

2, whenn = 0

a + 1, whenn = 1

a, whenn ≥ 2

H_n(a,a) =

H_n+1(a, 2), whenn ≥ 1

H_n(a, −1) =^{[nb 2]}

0, whenn = 0, orn ≥ 4

a − 1, whenn = 1

−a, whenn = 2

⁠1/a⁠ , whenn = 3

H_n(2, 2) =

3, whenn = 0

4, whenn ≥ 1, easily demonstrable recursively.

One of the earliest discussions of hyperoperations was that of Albert Bennett in 1914, who developed some of the theory ofcommutative hyperoperations (see§ Commutative hyperoperations below).^[8] About 12 years later,Wilhelm Ackermann defined the function${\displaystyle \varphi (a,b,n)}$, which somewhat resembles the hyperoperation sequence.^[17]

In his 1947 paper,^[7]Reuben Goodstein introduced the specific sequence of operations that are now calledhyperoperations, and also suggested the Greek namestetration, pentation, etc., for the extended operations beyond exponentiation (because they correspond to the indices 4, 5, etc.). As a three-argument function, e.g.,${\displaystyle G(n,a,b)=H_n(a,b)}$, the hyperoperation sequence as a whole is seen to be a version of the originalAckermann function${\displaystyle \varphi (a,b,n)}$ —recursive but notprimitive recursive — as modified by Goodstein to incorporate the primitivesuccessor function together with the other three basic operations of arithmetic (addition,multiplication,exponentiation), and to make a more seamless extension of these beyond exponentiation.

The original three-argumentAckermann function${\displaystyle \varphi }$ uses the same recursion rule as does Goodstein's version of it (i.e., the hyperoperation sequence), but differs from it in two ways. First,${\displaystyle \varphi (a,b,n)}$ defines a sequence of operations starting from addition (n = 0) rather than thesuccessor function, then multiplication (n = 1), exponentiation (n = 2), etc. Secondly, the initial conditions for${\displaystyle \varphi }$ result in${\displaystyle \varphi (a,b,3)=G(4,a,b+1)=a[4](b+1)}$, thus differing from the hyperoperations beyond exponentiation.^[9][18][19] The significance of theb + 1 in the previous expression is that${\displaystyle \varphi (a,b,3)}$ =${\displaystyle a^{a^{{\cdot }^{{\cdot }^{{\cdot }^a}}}}}$, whereb counts the number ofoperators (exponentiations), rather than counting the number ofoperands ("a"s) as does theb in${\displaystyle a[4]b}$, and so on for the higher-level operations. (See theAckermann function article for details.)

This is a list of notations that have been used for hyperoperations.

Name	Notation equivalent to${\displaystyle H_n(a,b)}$	Comment
Knuth's up-arrow notation	${\displaystyle a{\uparrow }^{n-2}b}$	Used byKnuth[20] (forn ≥ 3), and found in several reference books.[21][22]
Hilbert's notation	${\displaystyle {\varphi }_n(a,b)}$	Used byDavid Hilbert.[23]
Goodstein's notation	${\displaystyle G(n,a,b)}$	Used byReuben Goodstein.[7]
OriginalAckermann function	${\displaystyle \begin{array}{c}\varphi (a,b,n-1)\text{ for }1\le n\le 3\\ \varphi (a,b-1,n-1)\text{ for }n\ge 4\end{array}}$	Used byWilhelm Ackermann (forn ≥ 1)[17]
Ackermann–Péter function	${\displaystyle A(n,b-3)+3\text{for }a=2}$	This corresponds to hyperoperations for base 2 (a = 2)
Nambiar's notation	${\displaystyle a{\otimes }^{n-1}b}$	Used by Nambiar (forn ≥ 1)[24]
Superscript notation	${\displaystyle a{}^{(n)}b}$	Used byRobert Munafo.[18]
Subscript notation (for lower hyperoperations)	${\displaystyle a{}_{(n)}b}$	Used for lower hyperoperations by Robert Munafo.[18]
Operator notation (for "extended operations")	${\displaystyle a{O}_{n-1}b}$	Used for lower hyperoperations byJohn Doner andAlfred Tarski (forn ≥ 1).[25]
Square bracket notation	${\displaystyle a[n]b}$	Used in many online forums; convenient forASCII.
Conway chained arrow notation	${\displaystyle a\to b\to (n-2)}$	Used byJohn Horton Conway (forn ≥ 3)

In 1928,Wilhelm Ackermann defined a 3-argument function${\displaystyle \varphi (a,b,n)}$ which gradually evolved into a 2-argument function known as theAckermann function. Theoriginal Ackermann function${\displaystyle \varphi }$ was less similar to modern hyperoperations, because his initial conditions start with${\displaystyle \varphi (a,0,n)=a}$ for alln > 2. Also he assigned addition ton = 0, multiplication ton = 1 and exponentiation ton = 2, so the initial conditions produce very different operations for tetration and beyond.

n	Operation	Comment
0	${\displaystyle F_0(a,b)=a+b}$
1	${\displaystyle F_1(a,b)=a\cdot b}$
2	${\displaystyle F_2(a,b)=a^b}$
3	${\displaystyle F_3(a,b)=a[4](b+1)}$	An offset form oftetration. The iteration of this operation is different than theiteration of tetration.
4	${\displaystyle F_4(a,b)=(x\mapsto a[4](x+1){)}^b(a)}$	Not to be confused with pentation.

Another initial condition that has been used is${\displaystyle A(0,b)=2b+1}$ (where the base is constant${\displaystyle a=2}$), due toRózsa Péter, which does not form a hyperoperation hierarchy.

In 1984, C. W. Clenshaw and F. W. J. Olver began the discussion of using hyperoperations to prevent computerfloating-point overflows.^[26]Since then, many other authors^[27][28][29] have renewed interest in the application of hyperoperations tofloating-point representation. (SinceH_n(a,b) are all defined forb = -1.) While discussingtetration, Clenshawet al. assumed the initial condition${\displaystyle F_n(a,0)=0}$, which makes yet another hyperoperation hierarchy. Just like in the previous variant, the fourth operation is very similar totetration, but offset by one.

n	Operation	Comment
0	${\displaystyle F_0(a,b)=b+1}$
1	${\displaystyle F_1(a,b)=a+b}$
2	${\displaystyle F_2(a,b)=a\cdot b=e^{\ln (a)+\ln (b)}}$
3	${\displaystyle F_3(a,b)=a^b}$
4	${\displaystyle F_4(a,b)=a[4](b-1)}$	An offset form oftetration. The iteration of this operation is much different than theiteration of tetration.
5	${\displaystyle F_5(a,b)={\left(x\mapsto a[4](x-1)\right)}^b(0)=0\text{ if }a>0}$	Not to be confused with pentation.

An alternative for these hyperoperations is obtained by evaluation from left to right.^[11] Since

define (with ° or subscript)

${\displaystyle a_{(n+1)}b={\left(a_{(n+1)}(b-1)\right)}_{(n)}a}$

with

This was extended to ordinal numbers by Doner and Tarski,^[30] by :

It follows from Definition 1(i), Corollary 2(ii), and Theorem 9, that, fora ≥ 2 andb ≥ 1, that^{[original research?]}

${\displaystyle a{O}_{n}b=a_{(n+1)}b}$

But this suffers a kind of collapse, failing to form the "power tower" traditionally expected of hyperoperators:^{[31][nb 4]}

${\displaystyle {\alpha }_{(4)}(1+\beta )={\alpha }^{\left({\alpha }^{\beta }\right)}.}$

If α ≥ 2 and γ ≥ 2,^{[25][Corollary 33(i)][nb 4]}

${\displaystyle {\alpha }_{(1+2\nu +1)}\beta \le {\alpha }_{(1+2\nu )}(1+3\alpha \beta ).}$

n	Operation	Comment
0	${\displaystyle F_0(a,b)=a+1}$	Increment, successor, zeration
1	${\displaystyle F_1(a,b)=a+b}$
2	${\displaystyle F_2(a,b)=a\cdot b}$
3	${\displaystyle F_3(a,b)=a^b}$
4	${\displaystyle F_4(a,b)=a^{\left(a^{(b-1)}\right)}}$	Not to be confused withtetration.
5	${\displaystyle F_5(a,b)={\left(x\mapsto x^{x^{(a-1)}}\right)}^{b-1}(a)}$	Not to be confused with pentation. Similar totetration.

Commutative hyperoperations were considered by Albert Bennett as early as 1914,^[8] which is possibly the earliest remark about any hyperoperation sequence. Commutative hyperoperations are defined by the recursion rule

${\displaystyle F_{n+1}(a,b)=\exp (F_n(\ln (a),\ln (b)))}$

which is symmetric ina andb, meaning all hyperoperations are commutative. This sequence does not containexponentiation, and so does not form a hyperoperation hierarchy.

n	Operation	Comment
0	${\displaystyle F_0(a,b)=\ln \left(e^a+e^b\right)}$	Smooth maximum (LogSumExp)
1	${\displaystyle F_1(a,b)=a+b}$
2	${\displaystyle F_2(a,b)=a\cdot b=e^{\ln (a)+\ln (b)}}$	This is due to theproperties of the logarithm.
3	${\displaystyle F_3(a,b)=a^{\ln (b)}=e^{\ln (a)\ln (b)}}$	In afinite field, this is theDiffie–Hellman key exchange operation.
4	${\displaystyle F_4(a,b)=e^{e^{\ln (\ln (a))\ln (\ln (b))}}}$	Not to be confused withtetration.

R. L. Goodstein^[7] used the sequence of hyperoperators to create systems of numeration for the nonnegative integers. The so-calledcomplete hereditary representation of integern, at levelk and baseb, can be expressed as follows using only the firstk hyperoperators and using as digits only 0, 1, ...,b − 1, together with the baseb itself:

• For 0 ≤n ≤b − 1,n is represented simply by the corresponding digit.
• Forn >b − 1, the representation ofn is found recursively, first representingn in the form

b [k]x_k [k − 1]x_{k − 1} [k - 2] ... [2]x₂ [1]x₁

wherex_k, ...,x₁ are the largest integers satisfying (in turn)

b [k]x_k ≤n

b [k]x_k [k − 1]x_{k − 1} ≤n

...

b [k]x_k [k − 1]x_{k − 1} [k - 2] ... [2]x₂ [1]x₁ ≤n

Anyx_i exceedingb − 1 is then re-expressed in the same manner, and so on, repeating this procedure until the resulting form contains only the digits 0, 1, ...,b − 1, together with the baseb.

Unnecessary parentheses can be avoided by giving higher-level operators higher precedence in the order of evaluation; thus,

level-1 representations have the form b [1] X, withX also of this form;

level-2 representations have the form b [2] X [1] Y, withX,Y also of this form;

level-3 representations have the form b [3] X [2] Y [1] Z, withX,Y,Z also of this form;

level-4 representations have the form b [4] X [3] Y [2] Z [1] W, withX,Y,Z,W also of this form;

and so on.

In this type of base-bhereditary representation, the base itself appears in the expressions, as well as "digits" from the set {0, 1, ...,b − 1}. This compares toordinary base-2 representation when the latter is written out in terms of the baseb; e.g., in ordinary base-2 notation, 6 = (110)₂ = 2 [3] 2 [2] 1 [1] 2 [3] 1 [2] 1 [1] 2 [3] 0 [2] 0, whereas the level-3 base-2 hereditary representation is 6 = 2 [3] (2 [3] 1 [2] 1 [1] 0) [2] 1 [1] (2 [3] 1 [2] 1 [1] 0). The hereditary representations can be abbreviated by omitting any instances of [1] 0, [2] 1, [3] 1, [4] 1, etc.; for example, the above level-3 base-2 representation of 6 abbreviates to 2 [3] 2 [1] 2.

Examples:The unique base-2 representations of the number266, at levels 1, 2, 3, 4, and 5 are as follows:

Level 1: 266 = 2 [1] 2 [1] 2 [1] ... [1] 2 (with 133 2s)

Level 2: 266 = 2 [2] (2 [2] (2 [2] (2 [2] 2 [2] 2 [2] 2 [2] 2 [1] 1)) [1] 1)

Level 3: 266 = 2 [3] 2 [3] (2 [1] 1) [1] 2 [3] (2 [1] 1) [1] 2

Level 4: 266 = 2 [4] (2 [1] 1) [3] 2 [1] 2 [4] 2 [2] 2 [1] 2

Level 5: 266 = 2 [5] 2 [4] 2 [1] 2 [5] 2 [2] 2 [1] 2

The definitions of the hyperoperation sequence can naturally be transposed toterm rewriting systems (TRS).

The basic definition of the hyperoperation sequence corresponds with the reduction rules

To compute${\displaystyle H_n(a,b)}$ one can use astack, which initially contains the elements${\displaystyle ⟨n,a,b⟩}$.

Then, repeatedly until no longer possible, three elements are popped and replaced according to the rules^{[nb 5]}

Schematically, starting from${\displaystyle ⟨n,a,b⟩}$:

WHILE stackLength <> 1{POP 3 elements;PUSH 1 or 5 elements according to the rules r1, r2, r3, r4, r5;}

Example

Compute${\displaystyle H_2(2,2){\to }_{\ast }4}$.^[32]

The reduction sequence is^{[nb 5][nb 6]}

${\displaystyle \underset{\_}{H(S(S(0)),S(S(0)),S(S(0)))}}$

${\displaystyle {\to }_{r5}H(S(0),S(S(0)),\underset{\_}{H(S(S(0)),S(S(0)),S(0))})}$

${\displaystyle {\to }_{r5}H(S(0),S(S(0)),H(S(0),S(S(0)),\underset{\_}{H(S(S(0)),S(S(0)),0)}))}$

${\displaystyle {\to }_{r3}H(S(0),S(S(0)),\underset{\_}{H(S(0),S(S(0)),0)})}$

${\displaystyle {\to }_{r2}\underset{\_}{H(S(0),S(S(0)),S(S(0)))}}$

${\displaystyle {\to }_{r5}H(0,S(S(0)),\underset{\_}{H(S(0),S(S(0)),S(0))})}$

${\displaystyle {\to }_{r5}H(0,S(S(0)),H(0,S(S(0)),\underset{\_}{H(S(0),S(S(0)),0)}))}$

${\displaystyle {\to }_{r2}H(0,S(S(0)),\underset{\_}{H(0,S(S(0)),S(S(0)))})}$

${\displaystyle {\to }_{r1}\underset{\_}{H(0,S(S(0)),S(S(S(0))))}}$

${\displaystyle {\to }_{r1}S(S(S(S(0))))}$

When implemented using a stack, on input${\displaystyle ⟨2,2,2⟩}$

the stack configurations	represent the equations
${\displaystyle \underset{\_}{2,2,2}}$	${\displaystyle H_2(2,2)}$
${\displaystyle {\to }_{r5}1,2,\underset{\_}{2,2,1}}$	${\displaystyle =H_1(2,H_2(2,1))}$
${\displaystyle {\to }_{r5}1,2,1,2,\underset{\_}{2,2,0}}$	${\displaystyle =H_1(2,H_1(2,H_2(2,0)))}$
${\displaystyle {\to }_{r3}1,2,\underset{\_}{1,2,0}}$	${\displaystyle =H_1(2,H_1(2,0))}$
${\displaystyle {\to }_{r2}\underset{\_}{1,2,2}}$	${\displaystyle =H_1(2,2)}$
${\displaystyle {\to }_{r5}0,2,\underset{\_}{1,2,1}}$	${\displaystyle =H_0(2,H_1(2,1))}$
${\displaystyle {\to }_{r5}0,2,0,2,\underset{\_}{1,2,0}}$	${\displaystyle =H_0(2,H_0(2,H_1(2,0)))}$
${\displaystyle {\to }_{r2}0,2,\underset{\_}{0,2,2}}$	${\displaystyle =H_0(2,H_0(2,2))}$
${\displaystyle {\to }_{r1}\underset{\_}{0,2,3}}$	${\displaystyle =H_0(2,3)}$
${\displaystyle {\to }_{r1}4}$	${\displaystyle =4}$

The definition using iteration leads to a different set of reduction rules

As iteration isassociative, instead of rule r11 one can define

Like in the previous section the computation of${\displaystyle H_n(a,b)=H_n^1(a,b)}$ can be implemented using a stack.

Initially the stack contains the four elements${\displaystyle ⟨1,n,a,b⟩}$.

Then, until termination, four elements are popped and replaced according to the rules^{[nb 5]}

Schematically, starting from${\displaystyle ⟨1,n,a,b⟩}$:

WHILE stackLength <> 1{POP 4 elements;PUSH 1 or 7 elements according to the rules r6, r7, r8, r9, r10, r11;}

Example

Compute${\displaystyle H_3(0,3){\to }_{\ast }0}$.

On input${\displaystyle ⟨1,3,0,3⟩}$ the successive stack configurations are

The corresponding equalities are

When reduction rule r11 is replaced by rule r12, the stack is transformed according to

The successive stack configurations will then be

The corresponding equalities are

Remarks

• ${\displaystyle H_3(0,3)=0}$ is a special case, see§ Special cases above.^{[nb 3]}
• The computation of${\displaystyle H_n(a,b)}$ according to the rules {r6 - r10, r11} is heavily recursive. The culprit is the order in which iteration is executed:${\displaystyle H^n(a,b)=H(a,H^{n-1}(a,b))}$. The first${\displaystyle H}$ disappears only after the whole sequence is unfolded. For instance,${\displaystyle H_4(2,4)}$ converges to 65536 in 2863311767 steps, the maximum depth of recursion^{[nb 7]} is 65534.
• The computation according to the rules {r6 - r10, r12} is more efficient in that respect. The implementation of iteration${\displaystyle H^n(a,b)}$ as${\displaystyle H^{n-1}(a,H(a,b))}$ mimics the repeated execution of a procedure H.^{[nb 8]} The depth of recursion, (n+1), matches the loop nesting.Meyer & Ritchie (1967) formalized this correspondence. The computation of${\displaystyle H_4(2,4)}$ according to the rules {r6-r10, r12} also needs 2863311767 steps to converge on 65536, but the maximum depth of recursion is only 5, as tetration is the 5th operator in the hyperoperation sequence.
• The considerations above concern the recursion depth only. Either way of iterating leads to the same number of reduction steps, involving the same rules (when the rules r11 and r12 are considered "the same"). As the example shows the reduction of${\displaystyle H_3(0,3)}$ converges in 9 steps: 1 X r7, 3 X r8, 1 X r9, 2 X r10, 2 X r11/r12. The modus iterandi only affects the order in which the reduction rules are applied.

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