Tetration
In mathematics, tetration (or hyper4) is the next hyperoperation after exponentiation, but before pentation. It is defined as iterated exponentiation. The word was coined by Reuben Louis Goodstein from tetra (four) and iteration. Tetration is used for the notation of very large numbers. The notation means , which is the application of exponentiation times.
Shown here are the first four hyperoperations with tetration being the fourth of these. Succession, the unary operation denoted by , takes and yields the number after ; this is the zeroth hyperoperation.

Addition

 n copies of 1 added to a.


Multiplication

 n copies of a combined by addition.


Exponentiation

 n copies of a combined by multiplication.

 Tetration

 n copies of a combined by exponentiation, righttoleft.

The above example is read as "the nth tetration of a". Each operation is defined by iterating the previous one. Tetration is not an elementary function, unlike addition, subtraction, multiplication, and division.
Here, succession (a' = a + 1) is the most basic operation; addition (a + n) is a primary operation, though for natural numbers it can be thought of as a chained succession of n successors of a; multiplication (an) is also a primary operation, though for natural numbers it can be thought of as a chained addition involving n numbers a; and exponentiation () can be thought of as a chained multiplication involving n numbers a. Analogously, tetration () can be thought of as a chained power involving n numbers a. The parameter a may be called the baseparameter in the following, while the parameter n in the following may be called the heightparameter (which is integral in the first approach but may be generalized to fractional, real and complex heights, see below).
Contents
Formal definition
For any positive real and nonnegative integer , we define by:
Terminology
There are many terms for tetration, each of which has some logic behind it, but some have not become commonly used for one reason or another. Here is a comparison of each term with its rationale and counterrationale.
 The term tetration, introduced by Goodstein in his 1947 paper Transfinite Ordinals in Recursive Number Theory^{[1]} (generalizing the recursive baserepresentation used in Goodstein's theorem to use higher operations), has gained dominance. It was also popularized in Rudy Rucker's Infinity and the Mind.
 The term superexponentiation was published by Bromer in his paper Superexponentiation in 1987.^{[2]} It was used earlier by Ed Nelson in his book Predicative Arithmetic, Princeton University Press, 1986.
 The term hyperpower^{[3]} is a natural combination of hyper and power, which aptly describes tetration. The problem lies in the meaning of hyper with respect to the hyperoperation sequence. When considering hyperoperations, the term hyper refers to all ranks, and the term super refers to rank 4, or tetration. So under these considerations hyperpower is misleading, since it is only referring to tetration.
 The term power tower^{[4]} is occasionally used, in the form "the power tower of order n" for . This is a misnomer, however, because tetration cannot be expressed with iterated power functions (see above), since it is an iterated exponential function.
 The term snap is occasionally used in informal contexts, in the form "a snap n" for . This term is not yet widely accepted, although it is used within select communities.^{[citation needed]} It is believed to be a reference to jounce, the fourth derivative of position in physics, as tetration is the fourth hyperoperation and jounce is also known as snap.
Owing in part to some shared terminology and similar notational symbolism, tetration is often confused with closely related functions and expressions. Here are a few related terms:
Form Terminology Tetration Iterated exponentials Nested exponentials (also towers) Infinite exponentials (also towers)
In the first two expressions a is the base, and the number of times a appears is the height (add one for x). In the third expression, n is the height, but each of the bases is different.
Care must be taken when referring to iterated exponentials, as it is common to call expressions of this form iterated exponentiation, which is ambiguous, as this can either mean iterated powers or iterated exponentials.
Notation
There are many different notation styles that can be used to express tetration (also known as hyper4; some of them can be used as well for hyper5, hyper6, and higher hyperoperations).
Name Form Description Rudy Rucker notation Used by Maurer [1901] and Goodstein [1947]; Rudy Rucker's book Infinity and the Mind popularized the notation. Knuth's uparrow notation Allows extension by putting more arrows, or, even more powerfully, an indexed arrow. Conway chained arrow notation Allows extension by increasing the number 2 (equivalent with the extensions above), but also, even more powerfully, by extending the chain Ackermann function Allows the special case to be written in terms of the Ackermann function. Iterated exponential notation Allows simple extension to iterated exponentials from initial values other than 1. Hooshmand notations^{[5]}
Hyperoperation notations
Allows extension by increasing the number 4; this gives the family of hyperoperations. Text notation a^^n
a[4]n
Since the uparrow is used identically to the caret ( ^
), tetration may be written as (^^
); convenient for ASCII.Bowers's operators {a,b,4}
{a,b,4,1}
a {4} b
One notation above uses iterated exponential notation; in general this is defined as follows:
 with n "a"s.
There are not as many notations for iterated exponentials, but here are a few:
Name Form Description Standard notation Euler coined the notation , and iteration notation has been around about as long. Knuth's uparrow notation Allows for superpowers and superexponential function by increasing the number of arrows; used in the article on large numbers. Text notation exp_a^n(x)
Based on standard notation; convenient for ASCII. J Notation x^^:(n1)x
Repeats the exponentiation. See J (programming language)^{[6]}
Examples
Because of the extremely fast growth of tetration, most values in the following table are too large to write in scientific notation. In these cases, iterated exponential notation is used to express them in base 10. The values containing a decimal point are approximate.
1 1 1 1 1 2 4 16 65,536 2^{65,536} or (2.00353 × 10^{19,728}) 3 27 7,625,597,484,987 (3.6 × 10^{12} digits) 4 256 1.34078 × 10^{154} (8.1 × 10^{153} digits) 5 3,125 1.91101 × 10^{2,184} (1.3 × 10^{2,184} digits) 6 46,656 2.65912 × 10^{36,305} (2.1 × 10^{36,305} digits) 7 823,543 3.75982 × 10^{695,974} (3.2 × 10^{695,974} digits) 8 16,777,216 6.01452 × 10^{15,151,335} (5.4 × 10^{15,151,335} digits) 9 387,420,489 4.28125 × 10^{369,693,099} (4.1 × 10^{369,693,099} digits) 10 10,000,000,000 10^{10,000,000,000} (10^{10,000,000,000} digits)
Properties
Tetration has several properties that are similar to exponentiation, as well as properties that are specific to the operation and are lost or gained from exponentiation. Because exponentiation does not commute, the product and power rules do not have an analogue with tetration; the statements and are not necessarily true for all cases.^{[7]}
However, tetration does follow a different property, in which for integers, . Along with a given identity for the first a, this property can in fact be shown to be a recursive definition for tetration.
When a number x and 10 are coprime, it is possible to compute the last m decimal digits of using Euler's theorem, for any integer m.
Exponential towers
As we can see from the definition, when evaluating tetration expressed as an "exponentiation tower", the exponentiation is done at the deepest level first (in the notation, at the apex). For example:
Note that exponentiation is not associative, and evaluating the expression in the opposite order will lead to a different answer:
Exponential towers must be evaluated from top to bottom (or right to left). Computer programmers refer to this choice as rightassociative.
Extensions
Tetration can be extended beyond the positive integers to different domains, including , complex functions such as , and an infinite . Unlike exponents, the more limited properties of tetration reduce the ability to extend tetration to other bases such as a rational .
Extension of domain for bases
Base zero
The exponential is not consistently defined. Thus, the tetrations are not clearly defined by the formula given earlier. However, is well defined, and exists:
Thus we could consistently define . This is equivalent to defining .
Under this extension, , so the rule from the original definition still holds.
Complex bases
Since complex numbers can be raised to powers, tetration can be applied to bases of the form z = a + bi (where a and b are real). For example, in ^{n}z with z = i, tetration is achieved by using the principal branch of the natural logarithm; using Euler's formula we get the relation:
This suggests a recursive definition for ^{(n+1)}i = a' + b'i given any ^{n}i = a + bi:
The following approximate values can be derived:
Approximate value  

i  
0.2079  
0.9472 + 0.3208i  
0.0501 + 0.6021i  
0.3872 + 0.0305i  
0.7823 + 0.5446i  
0.1426 + 0.4005i  
0.5198 + 0.1184i  
0.5686 + 0.6051i 
Solving the inverse relation, as in the previous section, yields the expected ^{0}i = 1 and ^{(−1)}i = 0, with negative values of n giving infinite results on the imaginary axis. Plotted in the complex plane, the entire sequence spirals to the limit 0.4383 + 0.3606i, which could be interpreted as the value where n is infinite.
Such tetration sequences have been studied since the time of Euler, but are poorly understood due to their chaotic behavior. Most published research historically has focused on the convergence of the infinitely iterated exponential function. Current research has greatly benefited by the advent of powerful computers with fractal and symbolic mathematics software. Much of what is known about tetration comes from general knowledge of complex dynamics and specific research of the exponential map.
Extensions of the domain for different "heights"
This Extensions of the domain for iteration possibly contains original research. (February 2016) (Learn how and when to remove this template message)

Infinite heights
Tetration can be extended to infinite heights;^{[8]} i.e., for certain a and n values in , there exists a well defined result for an infinite n. This is because for bases within a certain interval, tetration converges to a finite value as the height tends to infinity. For example, converges to 2, and can therefore be said to be equal to 2. The trend towards 2 can be seen by evaluating a small finite tower:
In general, the infinitely iterated exponential , defined as the limit of as n goes to infinity, converges for e^{−e} ≤ x ≤ e^{1/e}, roughly the interval from 0.066 to 1.44, a result shown by Leonhard Euler.^{[9]} The limit, should it exist, is a positive real solution of the equation y = x^{y}. Thus, x = y^{1/y}. The limit defining the infinite tetration of x fails to converge for x > e^{1/e} because the maximum of y^{1/y} is e^{1/e}.
This may be extended to complex numbers z with the definition:
where W represents Lambert's W function.
As the limit y = ^{∞}x (if existent, i.e. for e^{−e} < x < e^{1/e}) must satisfy x^{y} = y we see that x ↦ y = ^{∞}x is (the lower branch of) the inverse function of y ↦ x = y^{1/y}.
Negative heights
To preserve the original rule for tetration,
for negative values of , we can use a recursive relation to prove :
 Substituting 1 for k gives
 .^{[10]}
Smaller negative values cannot be well defined in this way. Substituting 2 for k in the same equation gives
which is not well defined. They can, however, sometimes be considered sets.^{[10]}
For , any definition of is consistent with the rule because
 for any .
Real heights
At this time there is no commonly accepted solution to the general problem of extending tetration to the real or complex values of . There have, however, been multiple approaches towards the issue, and different approaches are outlined below.
In general the problem is finding, for any real a > 0, a superexponential function over real x > −2 that satisfies
 for all real ^{[11]}
To find a more natural extension, one or more extra requirements are usually required. This is usually some collection of the following:
 A continuity requirement (usually just that is continuous in both variables for ).
 A differentiability requirement (can be once, twice, k times, or infinitely differentiable in x).
 A regularity requirement (implying twice differentiable in x) that:
 for all
The fourth requirement differs from author to author, and between approaches. There are two main approaches to extending tetration to real heights; one is based on the regularity requirement, and one is based on the differentiability requirement. These two approaches seem to be so different that they may not be reconciled, as they produce results inconsistent with each other.
When is defined for an interval of length one, the whole function easily follows for all x > −2.
Linear approximation for real heights
A linear approximation (solution to the continuity requirement, approximation to the differentiability requirement) is given by:
hence:
Approximation  Domain 

for −1 < x < 0  
for 0 < x < 1  
for 1 < x < 2 
and so on. However, it is only piecewise differentiable; at integer values of x the derivative is multiplied by . It is continuously differentiable for if and only if . For example, using these methods and
A main theorem in Hooshmand's paper^{[5]} states: Let . If is continuous and satisfies the conditions:
 is differentiable on
 is a nondecreasing or nonincreasing function on
then is uniquely determined through the equation
where denotes the fractional part of x and is the iterated function of the function .
The proof is that the second through fourth conditions trivially imply that f is a linear function on [−1, 0].
The linear approximation to natural tetration function is continuously differentiable, but its second derivative does not exist at integer values of its argument. Hooshmand derived another uniqueness theorem for it which states:
If is a continuous function that satisfies:
 is convex on
then . [Here is Hooshmand's name for the linear approximation to the natural tetration function.]
The proof is much the same as before; the recursion equation ensures that and then the convexity condition implies that is linear on (−1, 0).
Therefore, the linear approximation to natural tetration is the only solution of the equation and which is convex on . All other sufficientlydifferentiable solutions must have an inflection point on the interval (−1, 0).
Higher order approximations for real heights
Beyond linear approximations, a quadratic approximation (to the differentiability requirement) is given by:
which is differentiable for all , but not twice differentiable. For example, If this is the same as the linear approximation. Note that because of the way it is calculated, this function does not "cancel out", contrary to exponents, where . Namely,
 .
Just as there is a quadratic approximation, cubic approximations and methods for generalizing to approximations of degree n also exist, although they are much more unwieldy and too long to include.^{[12]}
Complex heights
It has now been proven^{[13]} that there exists a unique function F which is a solution of the equation F(z + 1) = exp(F(z)) and satisfies the additional conditions that F(0)=1 and F(z) approaches the fixed points of the logarithm (roughly 0.318 ± 1.337i) as z approaches ±i∞ and that F is holomorphic in the whole complex zplane, except the part of the real axis at z ≤ −2. This proof confirms a previous conjecture.^{[14]} The complex map of this function is shown in the figure at right. The proof also works for other bases besides e, as long as the base is bigger than . The complex double precision approximation of this function is available online.^{[citation needed]}
The requirement of the tetration being holomorphic is important for its uniqueness. Many functions can be constructed as
where and are real sequences which decay fast enough to provide the convergence of the series, at least at moderate values of .
The function S satisfies the tetration equations S(z + 1) = exp(S(z)), S(0) = 1, and if α_{n} and β_{n} approach 0 fast enough it will be analytic on a neighborhood of the positive real axis. However, if some elements of {α} or {β} are not zero, then function S has multitudes of additional singularities and cutlines in the complex plane, due to the exponential growth of sin and cos along the imaginary axis; the smaller the coefficients {α} and {β} are, the further away these singularities are from the real axis.
The extension of tetration into the complex plane is thus essential for the uniqueness; the realanalytic tetration is not unique.
Nonelementary recursiveness
Tetration (restricted to ) is not an elementary recursive function. We can easily prove by induction that for every elementary recursive function f, there is a constant c such that
We denote the right hand side by . Suppose on the contrary that tetration is elementary recursive. is also elementary recursive. By the above inequality, there is a constant c such that . By letting , we have that , a contradiction.
Inverse operations
Exponentiation has two inverse operations; roots and logarithms. Analogously, the inverses of tetration are often called the superroot, and the superlogarithm (In fact, all all hyperoperations greater than or equal to 3 have analogous inverses); e.g., in the function , the two inverses are the cube superroot of y and the super logarithm base y of x.
Superroot
The superroot is the inverse operation of tetration with respect to the base: if , then y is an nth super root of x ( or ).
For example,
so 2 is the 4th superroot of 65,536 and
so 3 is the 3rd superroot of 19,683.
Square superroot
The 2ndorder superroot, square superroot, or super square root has two equivalent notations, and . It is the inverse of and can be represented with the Lambert W function:^{[15]}
The function also illustrates the reflective nature of the root and logarithm functions as the equation below only holds true when :
Like square roots, the square superroot of x may not have a single solution. Unlike square roots, determining the number of square superroots of x may be difficult. In general, if , then x has two positive square superroots between 0 and 1; and if , then x has one positive square superroot greater than 1. If x is positive and less than it doesn't have any real square superroots, but the formula given above yields countably infinitely many complex ones for any finite x not equal to 1.^{[15]} The function has been used to determine the size of data clusters.^{[16]}
Other superroots
For each integer n > 2, the function ^{n}x is defined and increasing for x ≥ 1, and ^{n}1 = 1, so that the nth superroot of x, , exists for x ≥ 1.
However, if the linear approximation above is used, then if −1 < y ≤ 0, so cannot exist.
In the same way as the square superroot, terminology for other super roots can be based off of the normal roots: "cube superroots" can be expressed as ; the "4th superroot" can be expressed as ; and the "n^{th} superroot" is . Note that may not be uniquely defined, because there may be more than one n^{th} root. For example, x has a single (real) superroot if n is odd, and up to two if n is even.^{[citation needed]}
Just as with the extension of tetration to infinite heights, the superroot can be extended to , being welldefined if 1/e ≤ x ≤ e. Note that and thus that . Therefore, when it is well defined, and, unlike normal tetration, is an elementary function. For example, .
It follows from the Gelfond–Schneider theorem that superroot for any positive integer n is either integer or transcendental, and is either integer or irrational.^{[17]} It is still an open question whether irrational superroots are transcendental in the latter case.
Superlogarithm
Once a continuous increasing (in x) definition of tetration, ^{x}a, is selected, the corresponding superlogarithm or is defined for all real numbers x, and a > 1.
The function slog_{a} x satisfies:
Open questions
There are several open questions concerning tetration, particularly when concerning the relations between number systems such as integers and irrational numbers
 It is not known whether there is a positive integer n for which ^{n}π or ^{n}e is an integer. In particular, it is not known whether ^{4}π is an integer.
 It is not known whether ^{n}q is an integer for any positive integer n and positive noninteger rational q.^{[17]} Particularly, it is not known whether the positive root of the equation ^{4}x = 2 is a rational number.
See also
 Ackermann function
 Big O notation
 Double exponential function
 Hyperoperation
 Iterated logarithm
 Symmetric levelindex arithmetic
References
 ^ R. L. Goodstein (1947). "Transfinite ordinals in recursive number theory". Journal of Symbolic Logic. 12 (4): 123–129. doi:10.2307/2266486. JSTOR 2266486.
 ^ N. Bromer (1987). "Superexponentiation". Mathematics Magazine. 60 (3): 169–174. JSTOR 2689566.
 ^ J. F. MacDonnell (1989). "Somecritical points of the hyperpower function x x … {\displaystyle x^{x^{\dots }}} ". International Journal of Mathematical Education. 20 (2): 297–305. doi:10.1080/0020739890200210. MR 0994348.
 ^ Weisstein, Eric W. "Power Tower". MathWorld.
 ^ ^{a} ^{b} M. H. Hooshmand, (2006). "Ultra power and ultra exponential functions". Integral Transforms and Special Functions. 17 (8): 549–558. doi:10.1080/10652460500422247.
 ^ "Power Verb". J Vocabulary. J Software. Retrieved 28 October 2011.
 ^ Alexander Meiburg. (2014). Extension of Tetration Through the Product PowerTower Retrieved November 29, 2018
 ^ "Climbing the ladder of hyper operators: tetration". George Daccache. January 5, 2015. Retrieved 18 February 2016.
 ^ Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." Acta Acad. Scient. Petropol. 2, 29–51, 1783. Reprinted in Euler, L. Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae. Leipzig, Germany: Teubner, pp. 350–369, 1921. (facsimile)
 ^ ^{a} ^{b} M. Muller. What comes beyond exponentiation? Retrieved December 4, 2018
 ^ Trappmann, Henryk; Kouznetsov, Dmitrii (June 28, 2010). "5+ methods for real analytic tetration" (PDF). Retrieved 5 December 2018.
 ^ Andrew Robbins. Solving for the Analytic Piecewise Extension of Tetration and the Superlogarithm. The extensions are found in part two of the paper, "Beginning of Results".
 ^ W. Paulsen and S. Cowgill (March 2017). "Solving F ( z + 1 ) = b F ( z ) {\displaystyle F(z+1)=b^{F(z)}} in the complex plane" (PDF). Advances in Computational Mathematics. 43: 1–22. doi:10.1007/s1044401795241.
 ^ D. Kouznetsov (July 2009). "Solution of F ( z + 1 ) = exp ( F ( z ) ) {\displaystyle F(z+1)=\exp(F(z))} in complex z {\displaystyle z} plane" (PDF). Mathematics of Computation. 78 (267): 1647–1670. doi:10.1090/S0025571809021887.
 ^ ^{a} ^{b} Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; Knuth, D. E. (1996). "On the Lambert W function" (PostScript). Advances in Computational Mathematics. 5: 333. doi:10.1007/BF02124750.
 ^ Krishnam R. (2004), "Efficient SelfOrganization Of Large Wireless Sensor Networks"  Dissertation, BOSTON UNIVERSITY, COLLEGE OF ENGINEERING. pp.3740
 ^ ^{a} ^{b} Marshall, Ash J., and Tan, Yiren, "A rational number of the form aa with a irrational", Mathematical Gazette 96, March 2012, pp. 106109.
 Daniel Geisler, tetration.org
 Ioannis Galidakis, On extending hyper4 to nonintegers (undated, 2006 or earlier) (A simpler, easier to read review of the next reference)
 Ioannis Galidakis, On Extending hyper4 and Knuth's Uparrow Notation to the Reals (undated, 2006 or earlier).
 Robert Munafo, Extension of the hyper4 function to reals (An informal discussion about extending tetration to the real numbers.)
 Lode Vandevenne, Tetration of the Square Root of Two, (2004). (Attempt to extend tetration to real numbers.)
 Ioannis Galidakis, Mathematics, (Definitive list of references to tetration research. Lots of information on the Lambert W function, Riemann surfaces, and analytic continuation.)
 Galidakis, Ioannis and Weisstein, Eric W. Power Tower
 Joseph MacDonell, Some Critical Points of the Hyperpower Function.
 Dave L. Renfro, Web pages for infinitely iterated exponentials
 Knobel R (1981). "Exponentials Reiterated". American Mathematical Monthly. 88: 235–252.
 Hans Maurer, "Über die Funktion für ganzzahliges Argument (Abundanzen)." Mittheilungen der Mathematische Gesellschaft in Hamburg 4, (1901), p. 33–50. (Reference to usage of from Knobel's paper.)
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