Pentated
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, pentation (or hyper-5) is the fifth
hyperoperation In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called ''hyperoperations'' in this context) that starts with a unary operation (the successor function with ''n'' = 0). The sequence continues with th ...
. Pentation is defined to be repeated
tetration In mathematics, tetration (or hyper-4) is an operation (mathematics), operation based on iterated, or repeated, exponentiation. There is no standard mathematical notation, notation for tetration, though Knuth's up arrow notation \uparrow \upa ...
, similarly to how tetration is repeated
exponentiation In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication ...
, exponentiation is repeated
multiplication Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...
, and multiplication is repeated
addition Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol, +) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication, and Division (mathematics), divis ...
. The concept of "pentation" was named by English mathematician
Reuben Goodstein Reuben Louis Goodstein (15 December 1912 – 8 March 1985) was an English mathematician with an interest in the philosophy and teaching of mathematics. Education Goodstein was educated at St Paul's School in London. He received his Master's de ...
in 1947, when he came up with the naming scheme for hyperoperations. The number ''a'' pentated to the number ''b'' is defined as ''a'' tetrated to itself ''b - 1'' times. This may variously be denoted as a , a\uparrow\uparrow\uparrow b, a\uparrow^3 b, a\to b\to 3, or , depending on one's choice of notation. For example, 2 pentated to 2 is 2 tetrated to 2, or 2 raised to the power of 2, which is 2^2 = 4. As another example, 2 pentated to 3 is 2 tetrated to the result of 2 tetrated to 2. Since 2 tetrated to 2 is 4, 2 pentated to 3 is 2 tetrated to 4, which is 2^ = 65536. Based on this definition, pentation is only defined when ''a'' and ''b'' are both
positive integers In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
, though progress has been made to allow for any value of ''a''.


Definition

Pentation is the next
hyperoperation In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called ''hyperoperations'' in this context) that starts with a unary operation (the successor function with ''n'' = 0). The sequence continues with th ...
(infinite
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of arithmetic operations, based on the previous one each time) after
tetration In mathematics, tetration (or hyper-4) is an operation (mathematics), operation based on iterated, or repeated, exponentiation. There is no standard mathematical notation, notation for tetration, though Knuth's up arrow notation \uparrow \upa ...
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 In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication ...
. It is a
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, 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 In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called ''hyperoperations'' in this context) that starts with a unary operation (the successor function with ''n'' = 0). The sequence continues with th ...
notation for pentation and tetration, 2 means 2 to itself 2 times, or 2 2 ). This can then be reduced to 2 2^2)=2 =2^=2^=2^=65,536.


Etymology

The word "pentation" was coined by
Reuben Goodstein Reuben Louis Goodstein (15 December 1912 – 8 March 1985) was an English mathematician with an interest in the philosophy and teaching of mathematics. Education Goodstein was educated at St Paul's School in London. He received his Master's de ...
in 1947 from the roots
penta- Numeral or number prefixes are prefixes derived from numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example: *triangle, quadrilateral, pentagon, hexagon, octagon ...
(five) and
iteration Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
. It is part of his general naming scheme for
hyperoperation In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called ''hyperoperations'' in this context) that starts with a unary operation (the successor function with ''n'' = 0). The sequence continues with th ...
s.


Notation

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 In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called ''hyperoperations'' in this context) that starts with a unary operation (the successor function with ''n'' = 0). The sequence continues with th ...
as a . In this format, a may be interpreted as the result of repeatedly applying the function x\mapsto a , for b repetitions, starting from the number 1. Analogously, a , tetration, represents the value obtained by repeatedly applying the function x\mapsto a , for b repetitions, starting from the number 1, and the pentation a represents the value obtained by repeatedly applying the function x\mapsto a , for b repetitions, starting from the number 1. This will be the notation used in the rest of the article. *In
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. In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called ''hyperoperatio ...
, a is represented as a \uparrow \uparrow \uparrow b or a \uparrow^b. In this notation, a\uparrow b represents the exponentiation function a^b and a\uparrow \uparrow b represents tetration. The operation can be easily adapted for hexation by adding another arrow. *In
Conway chained arrow notation Conway chained arrow notation, created by mathematician John Horton Conway, is a means of expressing certain extremely large numbers. It is simply a finite sequence of positive integers separated by rightward arrows, e.g. 2\to3\to4\to5\to6. As wi ...
, a = a\rightarrow b\rightarrow 3. *Another proposed notation is , though this is not extensible to higher hyperoperations.


Examples

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 In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total function, total computable function that is not Primitive recursive function, primitive recursive. ...
: 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 =A(4,b).. Although tetration, its base operation, was extended to non-integer heights (more generally any height, real or complex) in 2017, building on Hellmuth Kneser's work, pentation a is currently only defined for integer values of ''b'' where ''a'' > 0 and ''b'' ≥ −2, such as e = e \approx 2075.96834.... As with all hyperoperations of order 3 (
exponentiation In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication ...
) and higher, pentation has the following trivial cases (identities) which holds for all values of ''a'' and ''b'' within its domain: * 1 = 1 * a = a Additionally, we can also introduce the following defining relations: * a = a * a = 1 * a -1) = 0 * a -2) = -1 * a b+1) = a a ) 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 \exp_(n) = 10^n . * 2 = 2 = 2^2 = 4 * 2 = 2 2 ) = 2 2 ) = 2 = 2^ = 2^ = 2^ = 65,536 * 2 = 2 2 ) = 2 2 2 )) = 2 2 ) = 2 5,536 = 2^ \mbox \approx \exp_^(4.29508) * 2 = 2 2 ) = 2 2 2 2 ))) = 2 2 2 )) = 2 2 5,536) = 2^ \mbox \approx \exp_^(4.29508) * 3 = 3 = 3^ = 3^ = 7,625,597,484,987 * 3 = 3 3 ) = 3 3 ) = 3 ,625,597,484,987 = 3^ \mbox \approx \exp_^(1.09902) * 3 = 3 3 ) = 3 3 3 )) = 3 3 ,625,597,484,987) = 3^ \mbox \approx \exp_^(1.09902) * 4 = 4 = 4^ = 4^ \approx \exp_^3(2.19) (a number with over 10153 digits) * 5 = 5 = 5^ = 5^ \approx \exp_^4(3.33928) (a number with more than 10102184 digits)


See also

*
Ackermann function In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total function, total computable function that is not Primitive recursive function, primitive recursive. ...
*
Large numbers Large numbers, far beyond those encountered in everyday life—such as simple counting or financial transactions—play a crucial role in various domains. These expansive quantities appear prominently in mathematics, cosmology, cryptography, and s ...
*
Graham's number Graham's number is an Large numbers, immense number that arose as an upper bound on the answer of a problem in the mathematical field of Ramsey theory. It is much larger than many other large numbers such as Skewes's number and Moser's number, bot ...
*
History of large numbers Different cultures used different traditional numeral systems for naming large numbers. The extent of large numbers used varied in each culture. Two interesting points in using large numbers are the confusion on the term billion and milliard in ma ...


References

{{Large numbers Exponentials Large numbers Operations on numbers