mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though Knuth's...

now called hyperoperations. Goodstein also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation. The...

Reuben Goodstein after the Greek prefix of n suffixed with -ation (such as tetration (n = 4), pentation (n = 5), hexation (n = 6), etc.) and can be written...

fifth hyperoperation. Pentation is defined to be repeated tetration, similarly to how tetration is repeated exponentiation, exponentiation is repeated multiplication...

Wiktionary, the free dictionary. ^^ may refer to: A kaomoji Tetration, the ASCII form of the tetration operator ↑↑ Record separator, control character in the...

\uparrow 2\uparrow \uparrow (3+2\uparrow \uparrow 8),} which contains three tetrations. In 2019 this was further improved to: N ″ = ( 2 ↑↑ 5138 ) ⋅ ( ( 2 ↑↑...

in minor details of rounding) and forms an inverse to the operation of tetration. The iterated logarithm is useful in analysis of algorithms and computational...

= 2 × 2 {\displaystyle 2^{2}=2\times 2} . It is also equal to 32 (see tetration). The aliquot sum of 16 is 15, within an aliquot sequence of four composite...

by which the magnitude of a googolplex could be represented, such as tetration, hyperoperation, Knuth's up-arrow notation, Steinhaus–Moser notation,...

of two are common. The first 21 of them are: Also see Fermat number, Tetration and Hyperoperation § Lower hyperoperations. All of these numbers over...

number, larger than what can be represented even using power towers (tetration). However, it can be represented using layers of Knuth's up-arrow notation...

multiple up arrows, such as ⇈, for iterated, or repeated, exponentiation (tetration). The quantum theory of electron spin uses either upward or downward arrows...

exponentiation; this operation is sometimes called hyper-4 or tetration. Iterating tetration leads to another operation, and so on, a concept named hyperoperation...

functions, but much more slowly than double exponential functions. However, tetration and the Ackermann function grow faster. See Big O notation for a comparison...

} This sum is transcendental because it is a Liouville number. Like tetration, there is currently no accepted method of extension of the exponential...

which makes it a power of two. 256 is 4 raised to the 4th power, so in tetration notation, 256 is 24. 256 is the value of the expression n n {\displaystyle...

the function f ( x ) = x x . {\displaystyle f(x)=x^{x}.} The infinite tetration x x x ⋅ ⋅ ⋅ {\displaystyle x^{x^{x^{\cdot ^{\cdot ^{\cdot }}}}}} or ∞...

H-function Hyper operators Iterated logarithm Pentation Super-logarithms Tetration Lambert W function: Inverse of f(w) = w exp(w). Lamé function Mathieu...

logit. They are the inverse functions of the double exponential function, tetration, of f(w) = wew, and of the logistic function, respectively. From the perspective...

terms, is correct to the first decimal place when n is positive. Also see Tetration: f n(1) = n√2. Using the other fixed point a = f(4) = 4 causes the series...

function is a linear combination of its partial derivatives Euler's infinite tetration theorem – About the limit of iterated exponentiation Euler's rotation...

the set of real or rational numbers, is not commutative or associative. Tetration ( ↑↑ {\displaystyle \uparrow \uparrow } ), as a binary operation on the...

hyperoperations, used to build addition, multiplication, exponentiation, tetration, etc. It was studied in 1986 in an investigation involving generalization...

exponential time, though this term more commonly refers to time bounded by the tetration function. This complexity class can be characterized by a certain class...

studied by Hellmuth Kneser in 1950, later providing the basis for extending tetration to non-integer heights in 2017. The solutions of f(f(x)) = x over R {\displaystyle...

number is too large to reasonably express using exponentiation or even tetration. For more about modern usage for large numbers, see Large numbers. To...

{e^{e^{\cdot ^{\cdot ^{e}}}}} _{k\ e'{\text{s}}}=e\uparrow \uparrow k} using tetration or Knuth's up-arrow notation. To see the divergence of the series (4)...

more classes of growth behavior, like the hyperoperations beginning at tetration, and A ( n , n ) {\displaystyle A(n,n)} , the diagonal of the Ackermann...

in the OEIS). 65536 is 2 2 2 2 {\displaystyle 2^{2^{2^{2}}}} , so in tetration notation 65536 is 42. When expressed using Knuth's up-arrow notation,...

existence, uniqueness and their evaluation. The Ackermann functions and tetration can be interpreted in terms of superfunctions. Analysis of superfunctions...