Pentation

Pentation is the next hyperoperation (infinite sequence of arithmetic operations, based on the previous one each time) after tetration and before hexation. It is defined as iterated (repeated) tetration (assuming right-associativity). This is similar to tetration, as tetration is iterated right-associative exponentiation.{{citation

| last = Perstein | first = Millard H.

| date = June 1961

| doi = 10.1145/367766.368160

| issue = 6

| journal = Communications of the ACM

| page = 344

| title = Algorithm 93: General Order Arithmetic

| volume = 5| s2cid = 581764

| doi-access = free

}}. It is a binary operation defined with two numbers a and b, where a is tetrated to itself b − 1 times.

The type of hyperoperation is typically denoted by a number in brackets, []. For instance, using hyperoperation notation for pentation and tetration, $2[5]3$ means {{not a typo|tetrating}} 2 to itself 2 times, or $2[4](2[4]2)$. This can then be reduced to $2[4](2^2)=2[4]4=2^\{2^\{2^2\}\}=2^\{2^4\}=2^\{16\}=65,536$.

The word "pentation" was coined by Reuben Goodstein in 1947 from the roots penta- (five) and iteration. It is part of his general naming scheme for hyperoperations.{{citation

| last = Goodstein | first = R. L. | authorlink = Reuben Goodstein

| journal = The Journal of Symbolic Logic

| mr = 0022537

| pages = 123–129

| title = Transfinite ordinals in recursive number theory

| volume = 12

| issue = 4 | year = 1947| doi = 10.2307/2266486 | jstor = 2266486 | s2cid = 1318943 }}.

There is little consensus on the notation for pentation; as such, there are many different ways to write the operation. However, some are more used than others, and some have clear advantages or disadvantages compared to others.

• Pentation can be written as a hyperoperation as $a[5]b$. In this format, $a[3]b$ may be interpreted as the result of repeatedly applying the function $x\backslash mapsto\; a[2]x$, for $b$ repetitions, starting from the number 1. Analogously, $a[4]b$, tetration, represents the value obtained by repeatedly applying the function $x\backslash mapsto\; a[3]x$, for $b$ repetitions, starting from the number 1, and the pentation $a[5]b$ represents the value obtained by repeatedly applying the function $x\backslash mapsto\; a[4]x$, for $b$ repetitions, starting from the number 1.{{citation

| last = Knuth | first = D. E. | author-link = Donald Knuth

| doi = 10.1126/science.194.4271.1235

| journal = Science

| pages = 1235–1242

| title = Mathematics and computer science: Coping with finiteness

| volume = 194

| year = 1976

| pmid=17797067

| issue=4271| bibcode = 1976Sci...194.1235K | s2cid = 1690489 }}.{{citation

| last1 = Blakley | first1 = G. R.

| last2 = Borosh | first2 = I.

| doi = 10.1016/0001-8708(79)90052-5

| issue = 2

| journal = Advances in Mathematics

| mr = 549780

| pages = 109–136

| title = Knuth's iterated powers

| volume = 34

| year = 1979| doi-access =

}}. This will be the notation used in the rest of the article.

• In Knuth's up-arrow notation, $a[5]b$ is represented as $a\; \backslash uparrow\; \backslash uparrow\; \backslash uparrow\; b$ or $a\; \backslash uparrow^\{3\}b$. In this notation, $a\backslash uparrow\; b$ represents the exponentiation function $a^b$ and $a\backslash uparrow\; \backslash uparrow\; b$ represents tetration. The operation can be easily adapted for hexation by adding another arrow.
• In Conway chained arrow notation, $a[5]b\; =\; a\backslash rightarrow\; b\backslash rightarrow\; 3$.{{citation|title=The Book of Numbers|first1=John Horton|last1=Conway|author1-link=John Horton Conway|first2=Richard|last2=Guy|author2-link=Richard K. Guy|publisher=Springer|year=1996|isbn=9780387979939|page=61|url=https://books.google.com/books?id=0--3rcO7dMYC&pg=PA61}}.
• Another proposed notation is $\{\_\{b\}a\}$, though this is not extensible to higher hyperoperations.{{Cite web |title=Tetration.org - Tetration |url=http://www.tetration.org/Tetration/index.html |access-date=2022-09-12 |website=www.tetration.org}}

The values of the pentation function may also be obtained from the values in the fourth row of the table of values of a variant of the Ackermann function: if $A(n,m)$ is defined by the Ackermann recurrence $A(m-1,A(m,n-1))$ with the initial conditions $A(1,n)=an$ and $A(m,1)=a$, then $a[5]b=A(4,b)$.{{citation

| last = Nambiar | first = K. K.

| doi = 10.1016/0893-9659(95)00084-4

| issue = 6

| journal = Applied Mathematics Letters

| mr = 1368037

| pages = 51–53

| title = Ackermann functions and transfinite ordinals

| volume = 8

| year = 1995| doi-access = free

| citeseerx = 10.1.1.563.4668

}}.

Although tetration, its base operation, was extended to non-integer heights (more generally any height, real or complex) in 2017,{{cite journal |author-first1=W. |author-last1=Paulsen |author-first2=S. |author-last2=Cowgill |title=Solving $F(z+1)\; =\; b^\{F(z)\}$ in the complex plane |journal=Advances in Computational Mathematics |pages=1–22 |date=March 2017 |url=http://myweb.astate.edu/wpaulsen/tetration2.pdf |doi=10.1007/s10444-017-9524-1 |volume=43 |s2cid=9402035}} building on Hellmuth Kneser's work,{{cite journal |journal=Journal für die reine und angewandte Mathematik |author-first=H. |author-last=Kneser |language=de |title=Reelle analytische Lösungen der Gleichung $\backslash varphi\backslash Big(\backslash varphi(x)\backslash Big)=\{\backslash rm\; e\}^x$ und verwandter Funktionalgleichungen |volume=187 |pages=56–67 |date=1950}} pentation $a[5]b$ is currently only defined for integer values of b where a > 0 and b ≥ −2, such as $e[5]2\; =\; e[4]e\; \backslash approx\; 2075.96834...$. As with all hyperoperations of order 3 (exponentiation) and higher, pentation has the following trivial cases (identities) which holds for all values of a and b within its domain:

• $1[5]b\; =\; 1$
• $a[5]1\; =\; a$

Additionally, we can also introduce the following defining relations:

• $a[5]2\; =\; a[4]a$
• $a[5]0\; =\; 1$
• $a[5](-1)\; =\; 0$
• $a[5](-2)\; =\; -1$
• $a[5](b+1)\; =\; a[4](a[5]b)$

Other than the trivial cases shown above, pentation generates extremely large numbers very quickly. As a result, there are only a few non-trivial cases that produce numbers that can be written in conventional notation, which are all listed below.

Some of these numbers are written in power tower notation due to their extreme size. Note that $\backslash exp\_\{10\}(n)\; =\; 10^n$.

• $2[5]2\; =\; 2[4]2\; =\; 2^2\; =\; 4$
• $2[5]3\; =\; 2[4](2[5]2)\; =\; 2[4](2[4]2)\; =\; 2[4]4\; =\; 2^\{2^\{2^2\}\}\; =\; 2^\{2^4\}\; =\; 2^\{16\}\; =\; 65,536$
• $2[5]4\; =\; 2[4](2[5]3)\; =\; 2[4](2[4](2[4]2))\; =\; 2[4](2[4]4)\; =\; 2[4]65,536\; =\; 2^\{2^\{2^\{\backslash cdot^\{\backslash cdot^\{\backslash cdot^\{2\}\}\}\}\}\}\; \backslash mbox\{\; (a\; power\; tower\; of\; height\; 65,536)\; \}\; \backslash approx\; \backslash exp\_\{10\}^\{65,533\}(4.29508)$.
• $2[5]5\; =\; 2[4](2[5]4)\; =\; 2[4](2[4](2[4](2[4]2)))\; =\; 2[4](2[4](2[4]4))\; =\; 2[4](2[4]65,536)\; =\; 2^\{2^\{2^\{\backslash cdot^\{\backslash cdot^\{\backslash cdot^\{2\}\}\}\}\}\}\; \backslash mbox\{\; (a\; power\; tower\; of\; height\; 2[4]65,536)\; \}\; \backslash approx\; \backslash exp\_\{10\}^\{2[4]65,536-3\}(4.29508)$
• $3[5]2\; =\; 3[4]3\; =\; 3^\{3^3\}\; =\; 3^\{27\}\; =\; 7,625,597,484,987$
• $3[5]3\; =\; 3[4](3[5]2)\; =\; 3[4](3[4]3)\; =\; 3[4]7,625,597,484,987\; =\; 3^\{3^\{3^\{\backslash cdot^\{\backslash cdot^\{\backslash cdot^\{3\}\}\}\}\}\}\; \backslash mbox\{\; (a\; power\; tower\; of\; height\; 7,625,597,484,987)\; \}\; \backslash approx\; \backslash exp\_\{10\}^\{7,625,597,484,986\}(1.09902)$
• $3[5]4\; =\; 3[4](3[5]3)\; =\; 3[4](3[4](3[4]3))\; =\; 3[4](3[4]7,625,597,484,987)\; =\; 3^\{3^\{3^\{\backslash cdot^\{\backslash cdot^\{\backslash cdot^\{3\}\}\}\}\}\}\; \backslash mbox\{\; (a\; power\; tower\; of\; height\; 3[4]7,625,597,484,987)\; \}\; \backslash approx\; \backslash exp\_\{10\}^\{3[4]7,625,597,484,987-1\}(1.09902)$
• $4[5]2\; =\; 4[4]4\; =\; 4^\{4^\{4^4\}\}\; =\; 4^\{4^\{256\}\}\; \backslash approx\; \backslash exp\_\{10\}^3(2.19)$ (a number with over 10¹⁵³ digits)
• $5[5]2\; =\; 5[4]5\; =\; 5^\{5^\{5^\{5^5\}\}\}\; =\; 5^\{5^\{5^\{3125\}\}\}\; \backslash approx\; \backslash exp\_\{10\}^4(3.33928)$ (a number with more than 10¹⁰²¹⁸⁴ digits)

• Ackermann function
• Large numbers
• Graham's number
• History of large numbers

{{reflist}}

{{Hyperoperations}}

{{Large numbers}}

Category:Exponentials

Category:Large numbers

Category:Operations on numbers