Hyperoperation
In mathematics, the hyperoperation sequence[nb 1] is an infinite sequence of arithmetic operations (called hyperoperations in this context)[1][11][13] that starts with a unary operation (the successor function with n = 0). The sequence continues with the binary operations of addition (n = 1), multiplication (n = 2), and exponentiation (n = 3).
After that, the sequence proceeds with further binary operations extending beyond exponentiation, using right-associativity. For the operations beyond exponentiation, the nth member of this sequence is named by Reuben Goodstein after the Greek prefix of n suffixed with -ation (such as tetration (n = 4), pentation (n = 5), hexation (n = 6), etc.)[5] and can be written as using n − 2 arrows in Knuth's up-arrow notation.
Each hyperoperation may be understood recursively in terms of the previous one by:
- a[n]b=a[n−1](a[n−1](a[n−1](⋯[n−1](a[n−1](a[n−1]a))⋯)))⏟b copies of a,n≥2{displaystyle a[n]b=underbrace {a[n-1](a[n-1](a[n-1](cdots [n-1](a[n-1](a[n-1]a))cdots )))} _{displaystyle b{mbox{ copies of }}a},quad ngeq 2}
![{displaystyle a[n]b=underbrace {a[n-1](a[n-1](a[n-1](cdots [n-1](a[n-1](a[n-1]a))cdots )))} _{displaystyle b{mbox{ copies of }}a},quad ngeq 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b5b37b38e6798ec67d72e429a051aa7641fb571)
It may also be defined according to the recursion rule part of the definition, as in Knuth's up-arrow version of the Ackermann function:
- a[n]b=a[n−1](a[n](b−1)),n≥1{displaystyle a[n]b=a[n-1]left(a[n]left(b-1right)right),quad ngeq 1}
![{displaystyle a[n]b=a[n-1]left(a[n]left(b-1right)right),quad ngeq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acfe8d681527fde64d3bcacced77270f96900f25)
This can be used to easily show numbers much larger than those which scientific notation can, such as Skewes' number and googolplexplex (e.g. 50[50]50{displaystyle 50[50]50}![{displaystyle 50[50]50}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e849d9d2bf3c592596ad3b7670cb605f68ccf252)
This recursion rule is common to many variants of hyperoperations (see below in definition).
Contents
1 Definition
2 Examples
3 Special cases
4 History
5 Notations
5.1 Variant starting from a
5.2 Variant starting from 0
5.3 Lower hyperoperations
5.4 Commutative hyperoperations
6 See also
7 Notes
8 References
Definition
The hyperoperation sequence Hn(a,b):(N0)3→N0{displaystyle H_{n}(a,b),:,(mathbb {N} _{0})^{3}rightarrow mathbb {N} _{0}}
- Hn(a,b)=a[n]b={b+1if n=0aif n=1 and b=00if n=2 and b=01if n≥3 and b=0Hn−1(a,Hn(a,b−1))otherwise{displaystyle H_{n}(a,b)=a[n]b={begin{cases}b+1&{text{if }}n=0\a&{text{if }}n=1{text{ and }}b=0\0&{text{if }}n=2{text{ and }}b=0\1&{text{if }}ngeq 3{text{ and }}b=0\H_{n-1}(a,H_{n}(a,b-1))&{text{otherwise}}end{cases}}}
![{displaystyle H_{n}(a,b)=a[n]b={begin{cases}b+1&{text{if }}n=0\a&{text{if }}n=1{text{ and }}b=0\0&{text{if }}n=2{text{ and }}b=0\1&{text{if }}ngeq 3{text{ and }}b=0\H_{n-1}(a,H_{n}(a,b-1))&{text{otherwise}}end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d09394c6d619461dc93752b66df45b661a32a4cb)
(Note that for n = 0, the binary operation essentially reduces to a unary operation (successor function) by ignoring the first argument.)
For n = 0, 1, 2, 3, this definition reproduces the basic arithmetic operations of successor (which is a unary operation), addition, multiplication, and exponentiation, respectively, as
- H0(a,b)=b+1,H1(a,b)=a+b,H2(a,b)=a⋅b,H3(a,b)=ab,{displaystyle {begin{aligned}H_{0}(a,b)&=b+1,!,\H_{1}(a,b)&=a+b,!,\H_{2}(a,b)&=acdot b,!,\H_{3}(a,b)&=a^{b},!,end{aligned}}}
So what will be the next operation after exponentiation? We defined multiplication so that H2(a,3)=a[2]3=a×3=a+a+a,{displaystyle H_{2}(a,3)=a[2]3=atimes 3=a+a+a,}![{displaystyle H_{2}(a,3)=a[2]3=atimes 3=a+a+a,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83c8d39f8da198f717d8981669109e55dca411cc)
![{displaystyle H_{3}(a,3)=a[3]3=a^{3}=acdot acdot a,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e099408ee3d045c50580281b76add357b53a8e01)
![{displaystyle H_{4}(a,3)=a[4]3=operatorname {tetration} (a,3)=a^{a^{a}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1598365465790ae769ba5993e70a525bd72dcf71)
The H operations for n ≥ 3 can be written in Knuth's up-arrow notation as
- H4(a,b)=a↑↑b,H5(a,b)=a↑↑↑b,…Hn(a,b)=a↑n−2b for n≥3,…{displaystyle {begin{aligned}H_{4}(a,b)&=auparrow uparrow {b},!,\H_{5}(a,b)&=auparrow uparrow uparrow {b},!,\ldots &\H_{n}(a,b)&=auparrow ^{n-2}b{text{ for }}ngeq 3,!,\ldots &\end{aligned}}}
Knuth's notation could be extended to negative indices ≥ −2 in such a way as to agree with the entire hyperoperation sequence, except for the lag in the indexing:
- Hn(a,b)=a↑n−2b for n≥0.{displaystyle H_{n}(a,b)=auparrow ^{n-2}b{text{ for }}ngeq 0.}
The hyperoperations can thus be seen as an answer to the question "what's next" in the sequence: successor, addition, multiplication, exponentiation, and so on. Noting that
- a+b=(a+(b−1))+1a⋅b=a+(a⋅(b−1))ab=a⋅(a(b−1))a[4]b=aa[4](b−1){displaystyle {begin{aligned}a+b&=(a+(b-1))+1\acdot b&=a+(acdot (b-1))\a^{b}&=acdot left(a^{(b-1)}right)\a[4]b&=a^{a[4](b-1)}end{aligned}}}
![{displaystyle {begin{aligned}a+b&=(a+(b-1))+1\acdot b&=a+(acdot (b-1))\a^{b}&=acdot left(a^{(b-1)}right)\a[4]b&=a^{a[4](b-1)}end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5adbfca7ba2ea5ef0116db091a9fdd09f6bff1)
the relationship between basic arithmetic operations is illustrated, allowing the higher operations to be defined naturally as above. The parameters of the hyperoperation hierarchy are sometimes referred to by their analogous exponentiation term;[14] so a is the base, b is the exponent (or hyperexponent),[12] and n is the rank (or grade).[6], and Hn(a,b){displaystyle H_{n}(a,b)}
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 successor, addition, multiplication and exponentiation are all hyperoperations; the successor operation (producing x + 1 from x) is the most primitive, 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
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 | 1+b{displaystyle 1+b}  or a[0]b{displaystyle a[0]b}![{displaystyle a[0]b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e33bf19ab5a8ade49e2e4faad2899d7bd35bc5e) | 1+1+1+1+⋯+1+1+1⏟b copies of 1{displaystyle 1+underbrace {1+1+1+cdots +1+1+1} _{displaystyle b{mbox{ copies of 1}}}}  | hyper0, increment, successor, zeration | Arbitrary |
| 1 | a+b{displaystyle a+b}  or a[1]b{displaystyle a[1]b}![{displaystyle a[1]b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea88ed1e0ba5cc1204e06635803d66a81cab90ed) | a+1+1+1+⋯+1+1+1⏟b copies of 1{displaystyle a+underbrace {1+1+1+cdots +1+1+1} _{displaystyle b{mbox{ copies of 1}}}}  | hyper1, addition | Arbitrary |
| 2 | a⋅b{displaystyle acdot b}  or a[2]b{displaystyle a[2]b}![{displaystyle a[2]b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6761632b308609eeaf05407389355d3a98758c6f) | a+a+a+⋯+a+a+a⏟b copies of a{displaystyle underbrace {a+a+a+cdots +a+a+a} _{displaystyle b{mbox{ copies of }}a}}  | hyper2, multiplication | Arbitrary |
| 3 | ab{displaystyle a^{b}}  or a[3]b{displaystyle a[3]b}![{displaystyle a[3]b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd946603b3dd3c072053d13fbe120d44f70ac8ab) | a⋅a⋅a⋅⋯⋅a⋅a⋅a⏟b copies of a{displaystyle underbrace {acdot acdot acdot ;cdots ;cdot acdot acdot a} _{displaystyle b{mbox{ copies of }}a}}  | hyper3, exponentiation | b real, with some multivalued extensions to complex numbers |
| 4 | ba{displaystyle ^{b}a}  or a[4]b{displaystyle a[4]b}![{displaystyle a[4]b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfddc8a9d38692fe9ac6ecee1527c90dd2ecccca) | a[3](a[3](a[3](⋯[3](a[3](a[3]a))⋯)))⏟b copies of a{displaystyle underbrace {a[3](a[3](a[3](cdots [3](a[3](a[3]a))cdots )))} _{displaystyle b{mbox{ copies of }}a}} )cdots )))} _{displaystyle b{mbox{ copies of }}a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3612774b867e26b11d0e4653952cadba81128e4b) | hyper4, tetration | a ≥ 0 or an integer, b an integer ≥ −1[nb 2] (with some proposed extensions) |
| 5 | a[5]b{displaystyle a[5]b} ![{displaystyle a[5]b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d782af460e582816fb4e49d3907b621dd297d4d3) | a[4](a[4](a[4](⋯[4](a[4](a[4]a))⋯)))⏟b copies of a{displaystyle underbrace {a[4](a[4](a[4](cdots [4](a[4](a[4]a))cdots )))} _{displaystyle b{mbox{ copies of }}a}} )cdots )))} _{displaystyle b{mbox{ copies of }}a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae8020b2baedd1415289038c096c179cccb37794) | hyper5, pentation | a, b integers ≥ −1[nb 2] |
| 6 | a[6]b{displaystyle a[6]b} ![{displaystyle a[6]b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/440cac430b226ebceafac2548525b7f1aef2486c) | a[5](a[5](a[5](⋯[5](a[5](a[5]a))⋯)))⏟b copies of a{displaystyle underbrace {a[5](a[5](a[5](cdots [5](a[5](a[5]a))cdots )))} _{displaystyle b{mbox{ copies of }}a}} )cdots )))} _{displaystyle b{mbox{ copies of }}a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c85d0f485080c5075bc3918b09c9df863e5b3c0) | hyper6, hexation | a, b integers ≥ −1[nb 2] |
Special cases
Hn(0, b) =
- 0, when n = 2, or n = 3, b ≥ 1, or n ≥ 4, b odd (≥ −1)
- 1, when n = 3, b = 0, or n ≥ 4, b even (≥ 0)
b, when n = 1
b + 1, when n = 0
Hn(1, b) =
- 1, when n ≥ 3
b+1, when n ≤ 1
b, when n = 2
Hn(a, 0) =
- 0, when n = 2
- 1, when n = 0, or n ≥ 3
a, when n = 1
Hn(a, 1) =
a, when n ≥ 2- 2, when n = 0
a+1, when n = 1
Hn(a, a) =
Hn+1(a, 2), when n ≥ 1
a+1, when n = 0
Hn(a, −1) =[nb 2]
- 0, when n = 0, or n ≥ 4
a − 1, when n = 1- −a, when n = 2
1/a , when n = 3
Hn(2, 2) =
- 3, when n = 0
- 4, when n ≥ 1, easily demonstrable recursively.
History
One of the earliest discussions of hyperoperations was that of Albert Bennett[6] in 1914, who developed some of the theory of commutative hyperoperations (see below). About 12 years later, Wilhelm Ackermann defined the function ϕ(a,b,n){displaystyle phi (a,b,n)}
In his 1947 paper,[5]R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations, and also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation (because they correspond to the indices 4, 5, etc.). As a three-argument function, e.g., G(n,a,b)=Hn(a,b){displaystyle G(n,a,b)=H_{n}(a,b)}
The original three-argument Ackermann function ϕ{displaystyle phi }
}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3467af278839156bdfbae125af0a32d18cd902d6)
Notations
This is a list of notations that have been used for hyperoperations.
| Name | Notation equivalent to Hn(a,b){displaystyle H_{n}(a,b)} | Comment |
|---|---|---|
Knuth's up-arrow notation | a↑n−2b{displaystyle auparrow ^{n-2}b}  | Used by Knuth[18] (for n ≥ 3), and found in several reference books.[19][20] |
| Goodstein's notation | G(n,a,b){displaystyle G(n,a,b)}  | Used by Reuben Goodstein.[5] |
| Original Ackermann function | ϕ(a,b,n−1) for 1≤n≤3ϕ(a,b−1,n−1) for n≥4{displaystyle {begin{matrix}phi (a,b,n-1) {text{ for }}1leq nleq 3\phi (a,b-1,n-1) {text{ for }}ngeq 4end{matrix}}}  | Used by Wilhelm Ackermann (for n ≥ 1)[15] |
Ackermann–Péter function | A(n,b−3)+3 for a=2{displaystyle A(n,b-3)+3 {text{for }}a=2}  | This corresponds to hyperoperations for base 2 (a = 2) |
| Nambiar's notation | a⊗n−1b{displaystyle aotimes ^{n-1}b}  | Used by Nambiar (for n ≥ 1)[21] |
| Box notation | anb{displaystyle a{,{begin{array}{|c|}hline {!n!}\hline end{array}},}b}  | Used by Rubtsov and Romerio.[13][14] |
| Superscript notation | a(n)b{displaystyle a{}^{(n)}b}  | Used by Robert Munafo.[10] |
| Subscript notation (for lower hyperoperations) | a(n)b{displaystyle a{}_{(n)}b}  | Used for lower hyperoperations by Robert Munafo.[10] |
| Operator notation (for "extended operations") | aOn−1b{displaystyle aO_{n-1}b}  | Used for lower hyperoperations by John Donner and Alfred Tarski (for n ≥ 1).[22] |
| Square bracket notation | a[n]b{displaystyle a[n]b} ![{displaystyle a[n]b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2551ef8c0122ea93c537b613b561246ca3f8724f) | Used in many online forums; convenient for ASCII. |
Conway chained arrow notation | a→b→(n−2){displaystyle ato bto (n-2)}  | Used by John Horton Conway (for n ≥ 3) |
Bowers's operators | {a,b,n,1}{displaystyle {a,b,n,1}}  | Used by Jonathan Bowers (for n ≥ 1) |
Variant starting from a
In 1928, Wilhelm Ackermann defined a 3-argument function ϕ(a,b,n){displaystyle phi (a,b,n)}
n | Operation | Comment |
|---|---|---|
| 0 | F0(a,b)=a+b{displaystyle F_{0}(a,b)=a+b}  | |
| 1 | F1(a,b)=a⋅b{displaystyle F_{1}(a,b)=acdot b}  | |
| 2 | F2(a,b)=ab{displaystyle F_{2}(a,b)=a^{b}}  | |
| 3 | F3(a,b)=a[4](b+1){displaystyle F_{3}(a,b)=a[4](b+1)} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4cd80f3850f8a52ab3a1b23acba831f7148e692) | An offset form of tetration. The iteration of this operation is different than the iteration of tetration. |
| 4 | F4(a,b)=(x↦a[4](x+1))b(a){displaystyle F_{4}(a,b)=(xmapsto a[4](x+1))^{b}(a)} )^{b}(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f27ed90acc3f9f63d6c22be4366b0f3def9ea920) | Not to be confused with pentation. |
Another initial condition that has been used is A(0,b)=2b+1{displaystyle A(0,b)=2b+1}
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 overflows.[23] Since then, many other authors[24][25][26] have renewed interest in the application of hyperoperations to floating-point representation. (Since Hn(a, b) are all defined for b = -1.) While discussing tetration, Clenshaw et al. assumed the initial condition Fn(a,0)=0{displaystyle F_{n}(a,0)=0}
n | Operation | Comment |
|---|---|---|
| 0 | F0(a,b)=b+1{displaystyle F_{0}(a,b)=b+1}  | |
| 1 | F1(a,b)=a+b{displaystyle F_{1}(a,b)=a+b}  | |
| 2 | F2(a,b)=a⋅b=eln(a)+ln(b){displaystyle F_{2}(a,b)=acdot b=e^{ln(a)+ln(b)}}  | |
| 3 | F3(a,b)=ab{displaystyle F_{3}(a,b)=a^{b}}  | |
| 4 | F4(a,b)=a[4](b−1){displaystyle F_{4}(a,b)=a[4](b-1)} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/488ccddf037f628486ad2255768e6ddeb327ff23) | An offset form of tetration. The iteration of this operation is much different than the iteration of tetration. |
| 5 | F5(a,b)=(x↦a[4](x−1))b(0)=0 if a>0{displaystyle F_{5}(a,b)=left(xmapsto a[4](x-1)right)^{b}(0)=0{text{ if }}a>0} right)^{b}(0)=0{text{ if }}a>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96070b5541117aec6f43b67bc3269f9e8933f1f4) | Not to be confused with pentation. |
Lower hyperoperations
An alternative for these hyperoperations is obtained by evaluation from left to right. Since
- a+b=(a+(b−1))+1a⋅b=(a⋅(b−1))+aab=(a(b−1))⋅a{displaystyle {begin{aligned}a+b&=(a+(b-1))+1\acdot b&=(acdot (b-1))+a\a^{b}&=left(a^{(b-1)}right)cdot aend{aligned}}}
define (with ° or subscript)
- a(n+1)b=(a(n+1)(b−1))(n)a{displaystyle a_{(n+1)}b=left(a_{(n+1)}(b-1)right)_{(n)}a}
with
- a(1)b=a+ba(2)0=0a(n)1=afor n>2{displaystyle {begin{aligned}a_{(1)}b&=a+b\a_{(2)}0&=0\a_{(n)}1&=a&{text{for }}n>2\end{aligned}}}
This was extended to ordinal numbers by Donner and Tarski,[22][Definition 1] by :
- αO0β=α+βαOγβ=supη<β,ξ<γ(αOγη)Oξα{displaystyle {begin{aligned}alpha O_{0}beta &=alpha +beta \alpha O_{gamma }beta &=sup limits _{eta <beta ,xi <gamma }(alpha O_{gamma }eta )O_{xi }alpha end{aligned}}}
It follows from Definition 1(i), Corollary 2(ii), and Theorem 9, that, for a ≥ 2 and b ≥ 1, that[original research?]
- aOnb=a(n+1)b{displaystyle aO_{n}b=a_{(n+1)}b}
But this suffers a kind of collapse, failing to form the "power tower" traditionally expected of hyperoperators:[22][Theorem 3(iii)][nb 3]
- α(4)(1+β)=α(αβ).{displaystyle alpha _{(4)}(1+beta )=alpha ^{left(alpha ^{beta }right)}.}
If α ≥ 2 and γ ≥ 2,[22][Corollary 33(i)][nb 3]
- α(1+2γ+1)β≤α(1+2γ)(1+3αβ).{displaystyle alpha _{(1+2gamma +1)}beta leq alpha _{(1+2gamma )}(1+3alpha beta ).}
n | Operation | Comment |
|---|---|---|
| 0 | F0(a,b)=a+1{displaystyle F_{0}(a,b)=a+1}  | increment, successor, zeration |
| 1 | F1(a,b)=a+b{displaystyle F_{1}(a,b)=a+b} | |
| 2 | F2(a,b)=a⋅b{displaystyle F_{2}(a,b)=acdot b}  | |
| 3 | F3(a,b)=ab{displaystyle F_{3}(a,b)=a^{b}} | |
| 4 | F4(a,b)=a(a(b−1)){displaystyle F_{4}(a,b)=a^{left(a^{(b-1)}right)}}  | Not to be confused with tetration. |
| 5 | F5(a,b)=(x↦xx(a−1))b−1(a){displaystyle F_{5}(a,b)=left(xmapsto x^{x^{(a-1)}}right)^{b-1}(a)}  | Not to be confused with pentation. Similar to tetration. |
Commutative hyperoperations
Commutative hyperoperations were considered by Albert Bennett as early as 1914,[6] which is possibly the earliest remark about any hyperoperation sequence. Commutative hyperoperations are defined by the recursion rule
- Fn+1(a,b)=exp(Fn(ln(a),ln(b))){displaystyle F_{n+1}(a,b)=exp(F_{n}(ln(a),ln(b)))}
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.
n | Operation | Comment |
|---|---|---|
| 0 | F0(a,b)=ln(ea+eb){displaystyle F_{0}(a,b)=ln left(e^{a}+e^{b}right)}  | Smooth maximum |
| 1 | F1(a,b)=a+b{displaystyle F_{1}(a,b)=a+b} | |
| 2 | F2(a,b)=a⋅b=eln(a)+ln(b){displaystyle F_{2}(a,b)=acdot b=e^{ln(a)+ln(b)}} | This is due to the properties of the logarithm. |
| 3 | F3(a,b)=aln(b)=eln(a)ln(b){displaystyle F_{3}(a,b)=a^{ln(b)}=e^{ln(a)ln(b)}}  | A commutative form of exponentiation. |
| 4 | F4(a,b)=eeln(ln(a))ln(ln(b)){displaystyle F_{4}(a,b)=e^{e^{ln(ln(a))ln(ln(b))}}}  | Not to be confused with tetration. |
See also
- Large numbers
Notes
^ Sequences similar to the hyperoperation sequence have historically been referred to by many names, including: the Ackermann function[1] (3-argument), the Ackermann hierarchy,[2] the Grzegorczyk hierarchy[3][4] (which is more general), Goodstein's version of the Ackermann function,[5]operation of the nth grade,[6]z-fold iterated exponentiation of x with y,[7]arrow operations,[8]reihenalgebra[9] and hyper-n.[1][9][10][11][12]
^ abcd Let x = a[n](−1). By the recursive formula, a[n]0 = a[n − 1](a[n](−1)) ⇒ 1 = a[n − 1]x. One solution is x = 0, because a[n − 1]0 = 1 by definition when n ≥ 4. This solution is unique because a[n − 1]b > 1 for all a > 1, b > 0 (proof by recursion).
^ ab Ordinal addition is not commutative; see ordinal arithmetic for more information
References
^ abc Daniel Geisler (2003). "What lies beyond exponentiation?". Retrieved 2009-04-17..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
^ Harvey M. Friedman (Jul 2001). "Long Finite Sequences". Journal of Combinatorial Theory, Series A. 95 (1): 102–144. doi:10.1006/jcta.2000.3154. Archived from the original on 2013-02-02. Retrieved 2009-04-17.
^ Manuel Lameiras Campagnola and Cristopher Moore and José Félix Costa (Dec 2002). "Transfinite Ordinals in Recursive Number Theory". Journal of Complexity. 18 (4): 977–1000. doi:10.1006/jcom.2002.0655. Archived from the original on 2013-02-02. Retrieved 2009-04-17.
^ Marc Wirz (1999). "Characterizing the Grzegorczyk hierarchy by safe recursion". CiteSeer. Retrieved 2009-04-21.
^ abcd 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.
^ abcd Albert A. Bennett (Dec 1915). "Note on an Operation of the Third Grade". Annals of Mathematics. Second Series. 17 (2): 74–75. doi:10.2307/2007124. JSTOR 2007124.
^ ab Paul E. Black (2009-03-16). "Ackermann's function". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology (NIST). Retrieved 2009-04-17. External link in|work=(help)
^ J. E. Littlewood (Jul 1948). "Large Numbers". Mathematical Gazette. 32 (300): 163–171. doi:10.2307/3609933. JSTOR 3609933.
^ ab Markus Müller (1993). "Reihenalgebra" (PDF). Retrieved 2009-04-17.
^ abc Robert Munafo (November 1999). "Inventing New Operators and Functions". Large Numbers at MROB. Retrieved 2009-04-17.
^ ab A. J. Robbins (November 2005). "Home of Tetration". Archived from the original on 13 June 2015. Retrieved 2009-04-17.
^ ab I. N. Galidakis (2003). "Mathematics". Archived from the original on April 20, 2009. Retrieved 2009-04-17.
^ ab C. A. Rubtsov and G. F. Romerio (December 2005). "Ackermann's Function and New Arithmetical Operation". Retrieved 2009-04-17.
^ ab
G. F. Romerio (2008-01-21). "Hyperoperations Terminology". Tetration Forum. Retrieved 2009-04-21.
^ ab
Wilhelm Ackermann (1928). "Zum Hilbertschen Aufbau der reellen Zahlen". Mathematische Annalen. 99: 118–133. doi:10.1007/BF01459088.
^
Robert Munafo (1999-11-03). "Versions of Ackermann's Function". Large Numbers at MROB. Retrieved 2009-04-17.
^ J. Cowles and T. Bailey (1988-09-30). "Several Versions of Ackermann's Function". Dept. of Computer Science, University of Wyoming, Laramie, WY. Retrieved 2009-04-17.
^
Donald E. Knuth (Dec 1976). "Mathematics and Computer Science: Coping with Finiteness". Science. 194 (4271): 1235–1242. doi:10.1126/science.194.4271.1235. PMID 17797067. Retrieved 2009-04-21.
^
Daniel Zwillinger (2002). CRC standard mathematical tables and formulae, 31st Edition. CRC Press. p. 4. ISBN 1-58488-291-3.
^ Eric W. Weisstein (2003). CRC concise encyclopedia of mathematics, 2nd Edition. CRC Press. pp. 127–128. ISBN 1-58488-347-2.
^
K. K. Nambiar (1995). "Ackermann Functions and Transfinite Ordinals". Applied Mathematics Letters. 8 (6): 51–53. doi:10.1016/0893-9659(95)00084-4.
^ abcd John Donner; Alfred Tarski (1969). "An extended arithmetic of ordinal numbers". Fundamenta Mathematicae. 65: 95–127.
^
C.W. Clenshaw and F.W.J. Olver (Apr 1984). "Beyond floating point". Journal of the ACM. 31 (2): 319–328. doi:10.1145/62.322429. Retrieved 2009-04-21.
^
W. N. Holmes (Mar 1997). "Composite Arithmetic: Proposal for a New Standard". Computer. 30 (3): 65–73. doi:10.1109/2.573666. Retrieved 2009-04-21.
^
R. Zimmermann (1997). "Computer Arithmetic: Principles, Architectures, and VLSI Design" (PDF). Lecture notes, Integrated Systems Laboratory, ETH Zürich. Retrieved 2009-04-17.
^ T. Pinkiewicz and N. Holmes and T. Jamil (2000). "Design of a composite arithmetic unit for rational numbers". Proceedings of the IEEE. pp. 245–252. Retrieved 2009-04-17.
