# Tetration

Not to be confused with titration.
"Super-root" redirects here. For the directory supported by some Unixes, see super-root (computing).
Complex plot of the holomorphic tetration , with hue representing the function argument and brightness representing magnitude
, for n = 1, 2, 3 ..., showing convergence to the infinitely iterated exponential between the two dots

In mathematics, tetration (or hyper-4) 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):

n copies of 1 added to a.
2. Multiplication
n copies of a combined by addition.
3. Exponentiation
n copies of a combined by multiplication.
4. Tetration
n copies of a combined by exponentiation, right-to-left.

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 base-parameter in the following, while the parameter n in the following may be called the height-parameter (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 non-negative 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 right-associative.

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 counter-rationale.

• The term tetration, introduced by Goodstein in his 1947 paper Transfinite Ordinals in Recursive Number Theory[2] (generalizing the recursive base-representation 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 hyper-4; some of them can be used as well for hyper-5, hyper-6, 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 up-arrow 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 up-arrow 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 up-arrow notation Allows for super-powers and super-exponential 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^^:(n-1)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 × 1019,728
3 27 7,625,597,484,987 (3.6 × 1012 digits)
4 256 1.34078 × 10154 (8.1 × 10153 digits)
5 3,125 1.91101 × 102,184 (1.3 × 102,184 digits)
6 46,656 2.65912 × 1036,305 (2.1 × 1036,305 digits)
7 823,543 3.75982 × 10695,974 (3.2 × 10695,974 digits)
8 16,777,216 6.01452 × 1015,151,335 (5.4 × 1015,151,335 digits)
9 387,420,489 (3.7 × 108 digits) (4.1 × 10369,693,099 digits)
10 10,000,000,000 1010,000,000,000 (1010,000,000,000 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

Tetration by period
Tetration by escape

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

of the infinitely iterated exponential converges for the bases
The function on the complex plane, showing the real-valued infinitely iterated exponential function (black curve)

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 ee  x  e1/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 = xy. Thus, x = y1/y. The limit defining the infinite tetration of x fails to converge for x > e1/e because the maximum of y1/y is e1/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 ee < x < e1/e) must satisfy xy = y we see that x  y = x is (the lower branch of) the inverse function of y  x = y1/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 super-exponential 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
using linear approximation.

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 sufficiently-differentiable 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 top-down (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

Drawing of the analytic extension of tetration to the complex plane. Levels and levels are shown with thick curves.

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 z-plane, 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 αn and βn 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 real-analytic tetration is not unique.

## Open questions

• It is not known whether there is a positive integer n for which nπ or ne is an integer. In particular, it is not known whether 4π is an integer.
• It is not known whether nq is an integer for any positive integer n and positive non-integer rational q.[13] Particularly, it is not known whether the positive root of the equation 4x = 2 is a rational number.

## Inverse operations

Exponentiation has two inverse operations; roots and logarithms. Analogously, the inverses of tetration are often called the super-root, and the super-logarithm.

### Super-root

The super-root 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 super-root of 65,536 and

so 3 is the 3rd super-root (or super cube root) of 7,625,597,484,987.

#### Square super-root

The graph y = .

The 2nd-order super-root, square super-root, 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 super-root of x may not have a single solution. Unlike square roots, determining the number of square super-roots of x may be difficult. In general, if , then x has two positive square super-roots between 0 and 1; and if , then x has one positive square super-root greater than 1. If x is positive and less than it doesn't have any real square super-roots, 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 super-roots

For each integer n > 2, the function nx is defined and increasing for x ≥ 1, and n1 = 1, so that the nth super-root of x, , exists for x ≥ 1.

However, if the linear approximation above is used, then if 1 < y  0, so cannot exist.

Other super-roots 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 super-root can be expressed as ; and it can therefore be said that the nth super-root is . Note that may not be uniquely defined, because there may be more than one nth root. For example, x has a single (real) super-root if n is odd, and up to two if n is even.

The super-root can be extended to , and this shows a link to the mathematical constant e as it is only well-defined if 1/exe (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 super-root 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 super-roots are transcendental in the latter case.

### Super-logarithm

Main article: Super-logarithm

Once a continuous increasing (in x) definition of tetration, xa, is selected, the corresponding super-logarithm sloga x is defined for all real numbers x, and a > 1.

The function sloga x satisfies:

## References

1. 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 non-elementary, and so is tetration.
2. R. L. Goodstein (1947). "Transfinite ordinals in recursive number theory". Journal of Symbolic Logic. 12 (4): 123–129. doi:10.2307/2266486. JSTOR 2266486.
3. N. Bromer (1987). "Superexponentiation". Mathematics Magazine. 60 (3): 169–174. JSTOR 2689566.
4. 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.
5. M. H. Hooshmand, (2006). "Ultra power and ultra exponential functions". Integral Transforms and Special Functions. 17 (8): 549–558. doi:10.1080/10652460500422247.
6. Ioannis Galidakis. On Extending hyper4 and Knuth’s Up-arrow Notation to the Reals.
7. "Power Verb". J Vocabulary. J Software. Retrieved 28 October 2011.
8. "Climbing the ladder of hyper operators: tetration". George Daccache. January 5, 2015. Retrieved 18 February 2016.
9. Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." Acta Acad. Scient. Petropol. 2, 2951, 1783. Reprinted in Euler, L. Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae. Leipzig, Germany: Teubner, pp. 350369, 1921. (facsimile)
10. D. Kouznetsov (July 2009). "Solution of in complex -plane" (PDF). Mathematics of Computation. 78 (267): 1647–1670. doi:10.1090/S0025-5718-09-02188-7.
11. Marshall, Ash J., and Tan, Yiren, "A rational number of the form aa with a irrational", Mathematical Gazette 96, March 2012, pp. 106-109.
12. Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; Knuth, D. E. (1996). "On the Lambert W function" . Advances in Computational Mathematics. 5: 333. doi:10.1007/BF02124750.
13. BOSTON UNIVERSITY COLLEGE OF ENGINEERING – EFFICIENT SELF-ORGANIZATION OF LARGE WIRELESS SENSOR NETWORKS
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