Tetration
In mathematics, tetration (or hyper4) is the next hyperoperation after exponentiation, and 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 , the application of exponentiation times.
Shown here are the first four hyperoperations, with tetration as the fourth (and succession, the unary operation denoted taking and yielding the number after , as the 0th):
 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 (the next operation in the sequence is pentation). Tetration is neither an elementary function nor an elementary recursive function.^{[1]}
Here, succession is the most basic operation; addition is a primary operation, though for natural numbers it can be thought of as a chained succession of n successors of a; multiplication () 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).
Definition
For any positive real and nonnegative integer , we define by:
Iterated powers vs. iterated exponentials
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 highest level). For example:
Note that exponentiation is not associative, so evaluating the expression in the other 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.
When a and 10 are coprime, we can compute the last m decimal digits of using Euler's theorem.
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^{[2]} (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.^{[3]} It was used earlier by Ed Nelson in his book Predicative Arithmetic, Princeton University Press, 1986.
 The term hyperpower^{[4]} 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^{[5]} 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.
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^{[6]} Hyperoperation notations
Allows extension by increasing the number 4; this gives the family of hyperoperations Text notation a^^n
Since the uparrow is used identically to the caret ( ^
), tetration may be written as (^^
); convenient for ASCII.Bowers' array notation {a,b,2}
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. Ioannis Galidakis' notation Allows for large expressions in the base.^{[7]} 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)^{[8]}
Examples
In the following table, most values are too large to write in scientific notation, so iterated exponential notation is employed to express them in base 10. The values containing a decimal point are approximate.
1 1 1 1 1 2 4 16 65,536 2.00353 × 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 (3.7 × 10^{8} digits) (4.1 × 10^{369,693,099} digits) 10 10,000,000,000 10^{10,000,000,000} (10^{10,000,000,000} digits)
Extensions
Tetration can be extended to define and other domains as well.
Extension of domain for bases
Extension to 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.
Extension to complex bases
Since complex numbers can be raised to powers, tetration can be applied to bases of the form , where . For example, where , tetration is achieved by using the principal branch of the natural logarithm, and using Euler's formula we get the relation:
This suggests a recursive definition for given any :
The following approximate values can be derived:
Solving the inverse relation as in the previous section, yields the expected and , with negative values of n giving infinite results on the imaginary axis. Plotted in the complex plane, the entire sequence spirals to the limit , 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 (iteration) "heights"
Extension to infinite heights
Tetration can be extended to infinite heights^{[9]} (n in ). 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.^{[10]} 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}.
(Limited) extension to negative heights
In order to preserve the original rule:
for negative values of we must use the recursive relation:
Thus:
However smaller negative values cannot be well defined in this way because
which is not well defined.
Note further that for any definition of is consistent with the rule because
 for any .
Extension to 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 . Various approaches are mentioned below.
In general the problem is finding, for any real a > 0, a superexponential function over real x > −2 that satisfies
 A fourth requirement that is usually one of:
 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 the extension to real heights
A linear approximation (solution to the continuity requirement, approximation to the differentiability requirement) is given by:
hence:
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 .
Examples:
A main theorem in Hooshmand's paper^{[6]} 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 the extension to real heights
A quadratic approximation (to the differentiability requirement) is given by:
which is differentiable for all , but not twice differentiable. If this is the same as the linear approximation.
Note that this function does not satisfy condition that tetration "cancels out" (for example as in raising to power: ), because it is calculated topdown (as explained in section Iterated powers above) namely:
 .
A cubic approximation and a method for generalizing to approximations of degree n are given at.^{[11]}
Extension to complex heights
There is a conjecture^{[12]} 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 function is shown in the figure at right. The complex double precision approximation of this function is available online.
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.
Open questions
 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.^{[13]} Particularly, it is not known whether the positive root of the equation ^{4}x = 2 is a rational number.
Inverse operations
Exponentiation has two inverse operations; roots and logarithms. Analogously, the inverses of tetration are often called the superroot, and the superlogarithm.
Superroot
The superroot is the inverse operation of tetration with respect to the base: if , then y is an nth super root of x. For example,
so 2 is the 4th superroot of 65,536 and
so 3 is the 3rd superroot (or super cube root) of 7,625,597,484,987.
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:^{[14]}
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.^{[14]} The function has been used to determine the size of data clusters.^{[15]}
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.
Other superroots are expressible under the same basis used with normal roots: super cube roots, the function that produces y when , can be expressed as ; the 4th superroot can be expressed as ; and it can therefore be said that 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.
The superroot can be extended to , and this shows a link to the mathematical constant e as it is only welldefined if 1/e ≤ x ≤ e (see extension of tetration to infinite heights). Note that and thus that . Therefore, when it is well defined, and thus it 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.^{[13]} But 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 slog_{a} x is defined for all real numbers x, and a > 1.
The function slog_{a} x satisfies:
See also
 Ackermann function
 Double exponential function
 Hyperoperation
 Iterated logarithm
 Symmetric levelindex arithmetic
References
 ↑ It is easy to prove that for every elementary function f, there is a constant c s.t. . We can show (by using diagonal argument and above fact) that is nonelementary, and so is tetration.
 ↑ 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 ". International Journal of Mathematical Education. 20 (2): 297–305. doi:10.1080/0020739890200210. MR 994348.
 ↑ Weisstein, Eric W. "Power Tower". MathWorld.
 1 2 M. H. Hooshmand, (2006). "Ultra power and ultra exponential functions". Integral Transforms and Special Functions. 17 (8): 549–558. doi:10.1080/10652460500422247.
 ↑ Ioannis Galidakis. On Extending hyper4 and Knuth’s Uparrow Notation to the Reals.
 ↑ "Power Verb". J Vocabulary. J Software. Retrieved 28 October 2011.
 ↑ "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)
 ↑ Andrew Robbins. Solving for the Analytic Piecewise Extension of Tetration and the Superlogarithm.
 ↑ D. Kouznetsov (July 2009). "Solution of in complex plane" (PDF). Mathematics of Computation. 78 (267): 1647–1670. doi:10.1090/S0025571809021887.
 1 2 Marshall, Ash J., and Tan, Yiren, "A rational number of the form a^{a} with a irrational", Mathematical Gazette 96, March 2012, pp. 106109.
 1 2 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.
 ↑ BOSTON UNIVERSITY COLLEGE OF ENGINEERING – EFFICIENT SELFORGANIZATION OF LARGE WIRELESS SENSOR NETWORKS
 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 (Compilation of entries from questions about tetration on sci.math.)
 R. Knobel. "Exponentials Reiterated." American Mathematical Monthly 88, (1981), p. 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.)
Wikimedia Commons has media related to tetration. 
External links
 Daniel Geisler's site on tetration
 Tetration Forum
 Tetration  TORI  Mizugadro, the research site by Dmitrii Kouznetsov
 Gottfried Helms' site on tetration