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수학에서 '''하이퍼 연산 수열''' (Hyperoperation sequence) 이란, '''하이퍼 연산'''이라 불리는 [[덧셈]], [[곱셈]], [[거듭제곱]]으로 시작하는 [[이항연산]] [[수열]]이다. 이 수열의 ''n''번째 하이퍼 연산은 ''n''의 그리스어 접두사에 접미사 ''-ation''을 붙인 단어로 불리며, <!--The nth member of this sequence is named by Reuben Goodstein after the Greek prefix of n suffixed with -ation--> [[크누스 윗화살표 표기법]]에서 (n-2)개의 화살표로 표기할 수 있다. Each


== hyperoperationDefinition is defined recursively in terms of the previous one, as is the case with==
{{Main|Knuth's up-arrow notation. The part of the definition that does this is the recursion rule of the Ackermann function: }}
 
:<math>a \uparrow^n b = a \uparrow^{n-1} \left(a \uparrow^n (b-1)\right)</math>.
The ''hyperoperation sequence'' is a [[sequence]] <math>H_n</math> of [[binary operation]]s on <math>\mathbb{N}</math>, [[Index set|indexed]] by <math>\mathbb{N}</math>, which starts with [[addition]] (''n''&nbsp;=&nbsp;1), [[multiplication]] (''n''&nbsp;=&nbsp;2) and [[exponentiation]] <math>(n=3)</math>.
The parameters of the hyperoperation hierarchy are referred to by their analogous exponentiation term<ref name=romerioTerms>
{{cite web
| author= G. F. Romerio
| title= Hyperoperations Terminology
| url= http://math.eretrandre.org/tetrationforum/attachment.php?aid=208
| publisher= [http://math.eretrandre.org/tetrationforum/ Tetration Forum]
| date= 2008-01-21
| accessdate=2009-04-21}}</ref>; so ''a'' is the '''''base''''', ''b'' is the '''''exponent'''''
(or ''hyperexponent''<ref name=galidakis />), and ''n'' is the '''''rank''''' (or ''grade''<ref name=bennett />).
 
Using [[Knuth's up-arrow notation|arrow notation]] we can defined hyperoperations as
:<math>
H_n(a, b) = a \uparrow^{n-2} b =
\begin{cases}
b + 1 & \text{if } n = 0 \\
a & \text{if } n = 1, b = 0 \\
0 & \text{if } n = 2, b = 0 \\
1 & \text{if } n \ge 3, b = 0 \\
H_{n-1}(a, H_n(a, b - 1)) & \text{otherwise}
\end{cases}
</math>
 
It can be seen as an answer to the question "what's next" in the [[sequence]]: [[addition]], [[multiplication]], [[exponentiation]], and so on. Noting that
* <math>a + b = 1 + (a + (b - 1))</math>
* <math>a \times b = a + (a \times (b - 1))</math>
* <math>a ^ b = a \times (a ^ {(b - 1)})</math>
this produces a natural relationship between the hyperoperations, and allows higher operations to be defined, which produce very large numbers from small inputs, as further explained in the separate article on '''[[tetration]]'''.
 
In common terms, the hyperoperations are ways of compounding numbers that increase in growth based on the iteration of the previous hyperoperation. The concepts of addition, multiplication and exponentiation are all hyperoperations; the addition operator specifies the number of times 1 is to be added to itself to produce a final value, multiplication specifies the number of times a number is to be added to itself, and exponentiation refers to the number of times a number is to be multiplied by itself.
 
== Examples ==
 
This is a list of the first six hyperoperations.
 
{| class="wikitable"
|-
! ''n''
! Operation
! Definition
! Names
! Domain
|-
! 0
| <math>b + 1</math>
| <math>{ 1 + {\underbrace{1 + 1 + 1 + \cdots + 1}_{b}} }</math>
| hyper0, increment, [[successor function|successor]]
| ''b'' arbitrary
|-
! 1
| <math>a + b</math>
| <math>{ a + {\underbrace{1 + 1 + 1 + \cdots + 1}_{b}} }</math>
| hyper1, [[addition]]
| arbitrary
|-
! 2
| <math>ab</math>
| <math>{ {\underbrace{a + a + a + \cdots + a}} \atop{b} }</math>
| hyper2, [[multiplication]]
| arbitrary
|-
! 3
| <math>a \uparrow b = a^b</math>
| <math>{ {\underbrace{a \cdot a \cdot a \cdot a \cdot \ldots \cdot a}} \atop{b} }</math>
| hyper3, [[exponentiation]]
| ''a'' > 0, ''b'' real, or ''a'' non-zero, ''b'' an integer, with some multivalued extensions to complex numbers
|-
! 4
| <math>a \uparrow\uparrow b</math>
| <math>{ {\underbrace{a \uparrow a \uparrow a \uparrow \cdots \uparrow a}} \atop{b} }</math>
| hyper4, [[tetration]]
| ''a'' > 0, ''b'' an integer ≥ −1 (with some proposed extensions)
|-
! 5
| <math>a \uparrow\uparrow\uparrow b</math> or <math>a \uparrow^3 b</math>
| <math>{ {\underbrace{a \uparrow\uparrow a \uparrow\uparrow \cdots \uparrow\uparrow a}} \atop{b} }</math>
| hyper5, [[pentation]]
| ''a'' and ''b'' integers, ''a'' > 0, ''b'' ≥ 0
|-
! 6
| <math>a \uparrow^4 b</math>
| <math>{ {\underbrace{a \uparrow^3 a \uparrow^3 \cdots \uparrow^3 a}} \atop{b} }</math>
| hyper6, hexation
| ''a'' and ''b'' integers, ''a'' > 0, ''b'' ≥ 0
|}
 
See also [[Knuth's up-arrow notation#Tables of values|Tables of values]].
 
== History ==
 
One of the earliest discussions of hyperoperations was that of Albert Bennett<ref name=bennett /> in 1914, who developed some of the theory of ''commutative hyperoperations'' (see [[#Commutative hyperoperations|below]]). About 12 years later, [[Wilhelm Ackermann]] defined the function
<math>\phi(a, b, n)</math><ref name=ackOrig>{{cite journal
| author=Wilhelm Ackermann
| journal=[[Mathematische Annalen]]
| title=''Zum Hilbertschen Aufbau der reellen Zahlen''
| year=1928 | volume=99 | pages=118-133
| doi=10.1007/BF01459088}}</ref>
which somewhat resembles the hyperoperation sequence.
The ''original'' [[Ackermann function]] with three arguments used the same recursion rule, but it differs from modern hyperoperations in at least two ways. First, it assigned addition to <math>n=0</math>, multiplication to <math>n=1</math> and exponentiation to <math>n=2</math>. Secondly, the initial conditions of <math>\phi</math> indicate that <math>\phi(a, b, 3) = a \uparrow\uparrow (b + 1)</math>,
which produces very different values than hyperoperations above exponentiation.<ref name=black/><ref>{{cite web
| author= Robert Munafo
| title= Versions of Ackermann's Function
| url= http://www.mrob.com/pub/math/ln-2deep.html
| work= Large Numbers at MROB
| date= 1999-11-03
| accessdate=2009-04-17}}</ref><ref>{{cite web
| author= J. Cowles and T. Bailey
| title= Several Versions of Ackermann's Function
| url= http://www.cs.utexas.edu/users/boyer/ftp/nqthm/nqthm-1992/examples/basic/peter.events
| publisher= Dept. of Computer Science, University of Wyoming, Laramie, WY
| date= 1988-09-30
| accessdate=2009-04-17}}</ref>
 
In 1947, [[Reuben Goodstein]]<ref name=goodstein /> defined the hyperoperation sequence as it is used today,{{Dubious|Disputed names|date=June 2009}} where he used the notation <math>G(n,a,b)</math> for what would be written as <math>a \uparrow^{n-2}b</math> in [[Knuth's up-arrow notation|arrow notation]].
In the 1947 paper, Goodstein introduced the names "[[tetration]]", "pentation", "hexation", etc., for the successive operators beyond [[exponentiation]].
 
== Notations ==
 
This is a list of notations that have been used for hyperoperations.
 
{| class="wikitable"
|-
! Name
! Notation
! Comment
|-
| '''standard [[Knuth's up-arrow notation|arrow notation]]'''
| <math>a \uparrow^{n-2} b = H_n(a, b)</math>
| Used by Knuth,<ref name=knuth>{{cite journal
| author= Donald E. Knuth
| title= Mathematics and Computer Science: Coping with Finiteness
| journal= Science
| year= 1976
| month= Dec.
| volume= 194
| issue= 4271
| pages= 1235-1242
| url= http://www.sciencemag.org/cgi/content/abstract/194/4271/1235
| accessdate=2009-04-21}}</ref> and found in several reference books.<ref>{{cite book
| author = Daniel Zwillinger
| title = CRC standard mathematical tables and formulae, 31st Edition
| publisher = CRC Press
| year = 2002
| pages= 4
| isbn = 1584882913 }}</ref><ref>{{cite book
| author = Eric W. Weisstein
| title = CRC concise encyclopedia of mathematics, 2nd Edition
| publisher = CRC Press
| year = 2003
| pages= 127-128
| isbn = 1584883472 }}</ref>
|-
| ''Goodstein's notation''
| <math>G(n, a, b)</math>
| Used by [[Reuben Goodstein]].<ref name=goodstein />
|-
| ''original [[Ackermann function]]''
| <math>A(a, b, n-1) = H_n(a, b)</math>
| This is not quite the same as hyperoperations.
|-
| ''modern [[Ackermann function]]''
| <math>A(n, b - 3) + 3 = H_n(2, b)</math>
| This the same as hyperoperations for base 2.
|-
| ''Nambiar's notation''
| <math>a \otimes^n b</math>
| Used by Nambiar<ref>{{cite journal
| author= K. K. Nambiar
| title= Ackermann Functions and Transfinite Ordinals
| journal= Applied Mathematics Letters
| year= 1995
| volume= 8
| issue= 6
| pages= 51-53
| url= http://www.sciencemag.org/cgi/content/abstract/194/4271/1235
| accessdate=2009-04-21}}</ref>
|-
| ''Box notation''
| <math>a {\,\begin{array}{|c|}\hline{\!n\!}\\\hline\end{array}\,} b</math>
| Used by Rubtsov and Romerio.<ref name=romerioAck/><ref name=romerioTerms/>
|-
| ''Superscript notation''
| <math>a {}^{(n)} b</math>
| Used by Robert Munafo.<ref name=munafo/>
|-
| ''Subscript notation''
| <math>a {}_{(n)} b</math>
| Used for lower hyperoperations by Robert Munafo.<ref name=munafo/>
|-
| ''ASCII notation''
| <tt>a [n] b</tt>
| Used in many online forums. Based on box notation.
|}
 
== Generalization ==
 
For different initial conditions or different recursion rules, very different operations can occur. Some mathematicians refer to all variants as examples of hyperoperations.
 
In the general sense, a '''hyperoperation hierarchy''' <math>(S,\,I,\,F)</math> is a [[Indexed family|family]] <math>(F_n)_{n \in I}</math> of [[binary operation]]s on <math>S</math>, [[Index set|indexed]] by a set <math>I</math>, such that there exists <math>i, j, k \in I</math> where
* <math>F_i(a, b) = a + b</math> ([[addition]]),
* <math>F_j(a, b) = ab</math> ([[multiplication]]), and
* <math>F_k(a, b) = a^b</math> ([[exponentiation]]).
Also, if the last condition is relaxed (i.e. there is no [[exponentiation]]), then we may also include the commutative hyperoperations, described below. Although one could list each hyperoperation explicitly, this is generally not the case. Most variants only include the [[successor function]] (or [[addition]]) in their definition, and redefine [[multiplication]] (and beyond) based on a single recursion rule that applies to all ranks. Since this is part of the definition of the hierarchy, and not a property of the hierarchy itself, it is difficult to define formally.
 
There are many possibilities for hyperoperations that are different from Goodstein's version. By using different initial conditions for <math>F_n(a, 0)</math> or <math>F_n(a, 1)</math>, the iterations of these conditions may produce different hyperoperations above exponentiation, while still corresponding to addition and multiplication. The modern definition of hyperoperations includes <math>F_n(a, 0) = 1</math> for all <math>n \ge 3</math>, whereas the variants below include <math>F_n(a, 0) = a</math>, and <math>F_n(a, 0) = 0</math>.
 
An open problem in hyperoperation research is whether the hyperoperation hierarchy <math>(\mathbb{N}, \mathbb{N}, F)</math> can be generalized to <math>(\mathbb{C}, \mathbb{C}, F)</math>, and whether <math>(\mathbb{C}, F_n)</math> forms a [[quasigroup]] (with restricted domains).
 
=== Variant starting from <math>a</math> ===
{{Main|Ackermann function}}
 
In 1928, [[Wilhelm Ackermann]] defined a 3-argument function <math>\phi(a, b, n)</math> which gradually evolved into a 2-argument function known as the [[Ackermann function]]. The ''original'' Ackermann function <math>\phi</math> was less similar to modern hyperoperations, because his initial conditions start with <math>\phi(a, 0, n) = a</math> for all <math>n > 2</math>. Also he assigned addition to <math>(n = 0)</math>, multiplication to <math>(n = 1)</math> and exponentiation to <math>(n = 2)</math>, so the initial conditions produce very different operations for tetration and beyond.
 
{| class="wikitable"
|-
! ''n''
! Operation
! Comment
|-
! 0
| <math>F_0(a, b) = a + b</math>
|
|-
! 1
| <math>F_1(a, b) = ab</math>
|
|-
! 2
| <math>F_2(a, b) = a^b</math>
|
|-
! 3
| <math>F_3(a, b) = a \uparrow\uparrow (b + 1)</math>
| An offset form of [[tetration]]. The iteration of this operation is much different than the [[Iterated function|iteration]] of [[tetration]].
|-
! 4
| <math>F_4(a, b) = (x \to a \uparrow\uparrow (x + 1))^b(a)</math>
| Not to be confused with [[pentation]].
|}
 
Another initial condition that has been used is <math>A(0, b) = 2 b + 1</math> (where the base is constant <math>a=2</math>), due to Rózsa Péter, which does not form a hyperoperation hierarchy.
 
=== Variant starting from 0 ===
In 1984, C. W. Clenshaw and F. W. J. Olver began the discussion of using hyperoperations to prevent computer [[Floating point|floating-point]]
overflows.<ref name=clenshawolver>{{cite journal
| author= C.W. Clenshaw and F.W.J. Olver
| title= Beyond floating point
| journal= Journal of the ACM
| year= 1984
| month= Apr.
| volume= 31
| issue= 2
| pages= 319-328
| url= http://portal.acm.org/citation.cfm?id=62.322429
| accessdate=2009-04-21}}</ref>
Since then, many other authors<ref name=holmes>{{cite journal
| author= W. N. Holmes
| title= Composite Arithmetic: Proposal for a New Standard
| journal= Computer
| year= 1997
| month= Mar.
| volume= 30
| issue= 3
| pages= 65-73
| url= http://portal.acm.org/citation.cfm?id=620661
| accessdate=2009-04-21}}</ref><ref>{{cite web
| author= R. Zimmermann
| title= Computer Arithmetic: Principles, Architectures, and VLSI Design
| url= http://www.iis.ee.ethz.ch/~zimmi/publications/comp_arith_notes.pdf
| publisher= Lecture notes, Integrated Systems Laboratory, ETH Zürich
| date= 1997
| accessdate=2009-04-17}}</ref><ref>{{cite web
| author= T. Pinkiewicz and N. Holmes and T. Jamil
| title= Design of a composite arithmetic unit for rational numbers
| url= http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=845571
| publisher= Proceedings of the IEEE
| date= 2000
| pages= 245-252
| accessdate=2009-04-17}}</ref>
have renewed interest in the application of hyperoperations to [[Floating point|floating-point]] representation.
While discussion [[tetration]], Clenshaw ''et.al.'' assumed the initial condition <math>F_n(a, 0) = 0</math>, which makes yet another hyperoperation hierarchy. Just like in the previous variant, the fourth operation is very similar to [[tetration]], but offset by one.
 
{| class="wikitable"
|-
! ''n''
! Operation
! Comment
|-
! 1
| <math>F_1(a, b) = a + b</math>
|
|-
! 2
| <math>F_2(a, b) = ab = e^{\ln(a) + \ln(b)}</math>
|
|-
! 3
| <math>F_3(a, b) = a^b</math>
|
|-
! 4
| <math>F_4(a, b) = a \uparrow\uparrow (b - 1)</math>
| An offset form of [[tetration]]. The iteration of this operation is much different than the [[Iterated function|iteration]] of [[tetration]].
|-
! 5
| <math>F_5(a, b) = (x \to a \uparrow\uparrow (x - 1))^b(0)</math>
| Not to be confused with [[pentation]].
|}
 
=== Commutative hyperoperations ===
 
Commutative hyperoperations were considered by Albert Bennett as early as 1914,<ref name=bennett /> which is possibly the earliest remark about any hyperoperation sequence. Commutative hyperoperations are defined by the recursion rule
:<math>F_{n+1}(a, b) = \exp(F_n(\ln(a), \ln(b)))</math>
which is symmetric in ''a'' and ''b'', meaning all hyperoperations are commutative. This sequence does not contain [[exponentiation]], and so does not form a hyperoperation hierarchy.
 
{| class="wikitable"
|-
! ''n''
! Operation
! Comment
|-
! 0
| <math>F_0(a, b) = \ln(e^{a} + e^{b})</math>
|
|-
! 1
| <math>F_1(a, b) = a + b</math>
|
|-
! 2
| <math>F_2(a, b) = ab = e^{\ln(a) + \ln(b)}</math>
| This is due to the [[Logarithm#Properties of the logarithm|properties of the logarithm]].
|-
! 3
| <math>F_3(a, b) = e^{\ln(a)\ln(b)}</math>
| A [[Commutativity|commutative]] form of [[exponentiation]].
|-
! 4
| <math>F_4(a, b) = e^{e^{\ln(\ln(a))\ln(\ln(b))}}</math>
| Not to be confused with [[tetration]].
|}
 
=== Balanced hyperoperations ===
 
Balanced hyperoperations, first considered by Clément Frappier in 1991,<ref name=frappier>{{cite journal
| author= C. Frappier
| title= Iterations of a kind of exponentials
| journal= Fibonacci Quarterly
| year= 1991
| volume= 29
| issue= 4
| pages= 351–361}}</ref> are based on the iteration of the function <math>x^x</math>, and are thus related to [[Steinhaus-Moser notation]]. The recursion rule used in balanced hyperoperations is
:<math>F_{n+1}(a, b) = (x \to F_n(x, x))^{\log_2(b)}(a)</math>
which requires [[Continuous function|continuous]] [[iteration]], even for [[integer]] b.
 
{| class="wikitable"
|-
! ''n''
! Operation
! Comment
|-
! 0
|
| Rank 0 does not exist.<ref group="nb">
If there was a rank 0 balanced hyperoperation <math>f(a, b)</math>, then addition would be <math>a + b = (x \to f(x, x))^{\log_2(b)}(a)</math>. Substituting <math>b = 1</math> in this equation gives <math>a + 1 = (x \to f(x, x))^{0}(a) = a</math> which is a contradiction.
</ref>
|-
! 1
| <math>F_1(a, b) = a + b</math>
|
|-
! 2
| <math>F_2(a, b) = ab = a 2^{\log_2(b)}</math>
|
|-
! 3
| <math>F_3(a, b) = a^b = a^{2^{\log_2(b)}}</math>
| This is [[exponentiation]].
|-
! 4
| <math>F_4(a, b) = (x \to x^x)^{\log_2(b)}(a)</math>
| Not to be confused with [[tetration]].
|}
 
=== Lower hyperoperations ===
 
An alternative for these hyperoperations is obtained by evaluation from left to right. Since
* <math>a+b = (a+(b-1))+1</math>
* <math>a\times b = (a\times (b-1))+a</math>
* <math>a^b = (a^{(b-1)})\times a</math>
define (with subscripts instead of superscripts)
<math>a_{(n+1)}b = (a_{(n+1)}(b-1))_{(n)}a</math>
with
<math>a_{(1)}b = a+b</math>,
<math>a _ {(2)} 0 = 0</math>, and
<math>a _ {(n)} 0 = 1</math>
for
<math>n>2</math>
 
But this suffers a kind of collapse,
failing to form the "power tower" traditionally expected of hyper4:
<math>a_{(4)}b = a^{(a^{(b-1)})}</math>
 
How can <math>a^{(n)}b</math> be so different from <math>a_{(n)}b</math> for ''n>3''? This is because of a [[symmetry]] called [[associativity]] that's ''defined into'' + and × (see [[field (mathematics)|field]]) but which ^ lacks. It is more apt to say the two ''(n)''s were decreed to be the same for ''n<4''. (On the other hand, one can object that the field operations were defined to mimic what had been "observed in nature" and ask why "nature" suddenly objects to that symmetry…)
 
The other degrees do not collapse in this way, and so this family has some interest of its own as '''lower''' (perhaps '''lesser''' or '''inferior''') hyperoperations. With hyperfunctions greater than three, it is also '''lower''' in the sense that the answers you get are actually often a lot lower than the answers you get when using the standard method.
 
{| class="wikitable"
|-
! ''n''
! Operation
! Comment
|-
! 0
| <math>b + 1</math>
| increment, successor, zeration
|-
! 1
| <math>F_1(a, b) = a + b</math>
|
|-
! 2
| <math>F_2(a, b) = ab</math>
|
|-
! 3
| <math>F_3(a, b) = a^b</math>
| This is [[exponentiation]].
|-
! 4
| <math>F_4(a, b) = a^{a^{(b-1)}}</math>
| Not to be confused with [[tetration]].
|-
! 5
| <math>F_5(a, b) = (x \to x^{x^{(a-1)}})^{b-1}(a)</math>
| Not to be confused with pentation.
|}
 
==See also==
* [[Ackermann function]]
* [[Knuth's up-arrow notation]] (note the [[Knuth's up-arrow notation#Tables of values|Tables of values]])
* [[Conway chained arrow notation]]
* [[Tetration]]
* [[Large numbers]]
 
== Notes ==
<references group="nb" />
 
== References ==
{{reflist|refs=
 
<ref name=bennett>{{cite journal
| author= Albert A. Bennett
| title= Note on an Operation of the Third Grade
| journal= Annals of Mathematics, Second Series
| year= 1915
| month= Dec.
| volume= 17
| issue= 2
| pages= 74-75
| url= http://www.jstor.org/stable/2007124
| accessdate=2009-04-17}}</ref>,
 
<ref name=black>{{cite web
| author= Paul E. Black
| title= Ackermann's function
| url= http://www.itl.nist.gov/div897/sqg/dads/HTML/ackermann.html
| work= [http://www.itl.nist.gov/div897/sqg/dads/ Dictionary of Algorithms and Data Structures]
| publisher= U.S. National Institute of Standards and Technology (NIST)
| date= 2009-03-16
| accessdate=2009-04-17}}</ref>,
 
<ref name=campagnola>{{cite journal
| author= Manuel Lameiras Campagnola and Cristopher Moore and José Félix Costa
| title= Transfinite Ordinals in Recursive Number Theory
| journal= Journal of Complexity
| year= 2002
| month= Dec.
| volume= 18
| issue= 4
| pages= 977-1000
| url= http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHX-482XFM6-4&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=f4e3ffc2a28e8abd16cde236197fd487
| accessdate=2009-04-17}}</ref>
 
<ref name=friedman>{{cite journal
| author= Harvey M. Friedman
| title= Long Finite Sequences
| journal= Journal of Combinatorial Theory, Series A
| year= 2001
| month= Jul.
| volume= 95
| issue= 1
| pages= 102-144
| url= http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHS-45RFJ9C-5&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_version=1&_urlVersion=0&_userid=10&md5=8097ac57c9dbe05b99fef6a95309f1df
| accessdate=2009-04-17}}</ref>
 
<ref name=galidakis>{{cite web
| author= I. N. Galidakis
| title= Mathematics
| url= http://ioannis.virtualcomposer2000.com/math/index.html
| date= 2003
| accessdate=2009-04-17}}</ref>
 
<ref name=geisler>{{cite web
| author= Daniel Geisler
| title= What lies beyond exponentiation?
| url= http://tetration.org/
| date= 2003
| accessdate=2009-04-17}}</ref> (3-argument), the
 
<ref name=goodstein>{{cite journal
| author= R. L. Goodstein
| title= Transfinite Ordinals in Recursive Number Theory
| journal= Journal of Symbolic Logic
| year= 1947
| month= Dec.
| volume= 12
| issue= 4
| pages= 123-129
| url= http://www.jstor.org/stable/2266486
| accessdate=2009-04-17}}</ref>
 
<ref name=littlewood>{{cite journal
| author= J. E. Littlewood
| title= Large Numbers
| journal= Mathematical Gazette
| year= 1948
| month= Jul.
| volume= 32
| issue= 300
| pages= 163-171
| url= http://www.jstor.org/stable/3609933
| accessdate=2009-04-17}}</ref>,
 
<ref name=mueller>{{cite web
| author= Markus Müller
| title= Reihenalgebra
| url= http://www.math.tu-berlin.de/~mueller/reihenalgebra.pdf
| date= 1993
| accessdate=2009-04-17}}</ref>
 
<ref name=munafo>{{cite web
| author= Robert Munafo
| title= Inventing New Operators and Functions
| url= http://www.mrob.com/pub/math/largenum-3.html
| work= Large Numbers at MROB
| date= 1999-11
| accessdate=2009-04-17}}</ref>
 
<ref name=robbins>{{cite web
| author= A. J. Robbins
| title= Home of Tetration
| url= http://tetration.itgo.com/index.html
| date= 2005-11
| accessdate=2009-04-17}} {{Dead link|date=October 2009}}</ref>
 
<ref name=romerioAck>{{cite web
| author= C. A. Rubtsov and G. F. Romerio
| title= Ackermann's Function and New Arithmetical Operation
| url= http://forum.wolframscience.com/showthread.php?s=&threadid=956
| date= 2005-12
| accessdate=2009-04-17}}</ref>
 
<ref name=wirz>{{cite web
| author= Marc Wirz
| title= Characterizing the Grzegorczyk hierarchy by safe recursion
| url= http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.3374
| publisher= CiteSeer
| date= 1999
| accessdate=2009-04-21}}</ref>
 
<ref name=wolfram>{{cite book
| author = Stephen Wolfram
| title = A New Kind of Science (NKS)
| publisher = [[Wolfram Research|Wolfram Media]]
| year = 2002
| pages= 897?
| isbn = 1579550088 }}</ref>
 
}}
 
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[[Category:Large numbers]]
 
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[[ja:ハイパー演算子]]
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