Tetration
In mathematics, tetration (or hyper4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though Knuth's up arrow notation and the leftexponent ^{x}b are common.
Under the definition as repeated exponentiation, means , where n copies of a are iterated via exponentiation, righttoleft, i.e. the application of exponentiation times. n is called the "height" of the function, while a is called the "base," analogous to exponentiation. It would be read as "the nth tetration of a".
It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra (four) and iteration.
Tetration is also defined recursively as
allowing for attempts to extend tetration to nonnatural numbers such as real, complex, and ordinal numbers.
The two inverses of tetration are called superroot and superlogarithm, analogous to the nth root and the logarithmic functions. None of the three functions are elementary.
Tetration is used for the notation of very large numbers.
Introduction
[edit]The first four hyperoperations are shown here, with tetration being considered the fourth in the series. The unary operation succession, defined as , is considered to be the zeroth operation.
 Addition n copies of 1 added to a combined by succession.
 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.
Importantly, nested exponents are interpreted from the top down: means and not
Succession, , is the most basic operation; while addition () is a primary operation, for addition of natural numbers it can be thought of as a chained succession of successors of ; multiplication () is also a primary operation, though for natural numbers it can analogously be thought of as a chained addition involving numbers of . Exponentiation can be thought of as a chained multiplication involving numbers of and tetration () as a chained power involving numbers . Each of the operations above are defined by iterating the previous one;^{[1]} however, unlike the operations before it, tetration is not an elementary function.
The parameter is referred to as the base, while the parameter may be referred to as the height. In the original definition of tetration, the height parameter must be a natural number; for instance, it would be illogical to say "three raised to itself negative five times" or "four raised to itself one half of a time." However, just as addition, multiplication, and exponentiation can be defined in ways that allow for extensions to real and complex numbers, several attempts have been made to generalize tetration to negative numbers, real numbers, and complex numbers. One such way for doing so is using a recursive definition for tetration; for any positive real and nonnegative integer , we can define recursively as:^{[1]}
The recursive definition is equivalent to repeated exponentiation for natural heights; however, this definition allows for extensions to the other heights such as , , and as well – many of these extensions are areas of active research.
Terminology
[edit]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 . Exponentiation is easily misconstrued: note that the operation of raising to a power is rightassociative (see below). Tetration is iterated exponentiation (call this rightassociative operation ^), starting from the top right side of the expression with an instance a^a (call this value c). Exponentiating the next leftward a (call this the 'next base' b), is to work leftward after obtaining the new value b^c. Working to the left, use the next a to the left, as the base b, and evaluate the new b^c. 'Descend down the tower' in turn, with the new value for c on the next downward step.
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:
Terminology  Form 

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
[edit]There are many different notation styles that can be used to express tetration. Some notations can also be used to describe other hyperoperations, while some are limited to tetration and have no immediate extension.
Name  Form  Description 

Rudy Rucker notation  Used by Maurer [1901] and Goodstein [1947]; Rudy Rucker's book Infinity and the Mind popularized the notation.^{[nb 1]}  
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]}  Used by M. H. Hooshmand [2006].  
Hyperoperation notations  Allows extension by increasing the number 4; this gives the family of hyperoperations.  
Double caret notation  a^^n

Since the uparrow is used identically to the caret (^ ), tetration may be written as (^^ ); convenient for ASCII.

One notation above uses iterated exponential notation; this is defined in general as follows:
 with n as.
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)^{[7]} 
Infinity barrier notation  Jonathan Bowers coined this,^{[8]} and it can be extended to higher hyperoperations 
Examples
[edit]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  1  1 

2  4 (2^{2})  16 (2^{4})  65,536 (2^{16})  2.00353 × 10^{19,728}  (10^{6.03123×1019,727})  (10^{106.03123×1019,727}) 
3  27 (3^{3})  7,625,597,484,987 (3^{27})  (1.25801 × 10^{3,638,334,640,024} ^{[9]})  
4  256 (4^{4})  1.34078 × 10^{154} (4^{256})  (10^{8.0723×10153})  
5  3,125 (5^{5})  1.91101 × 10^{2,184} (5^{3,125})  (10^{1.33574×102,184})  
6  46,656 (6^{6})  2.65912 × 10^{36,305} (6^{46,656})  (10^{2.0692×1036,305})  
7  823,543 (7^{7})  3.75982 × 10^{695,974} (7^{823,543})  (3.17742 × 10^{695,974} digits)  
8  16,777,216 (8^{8})  6.01452 × 10^{15,151,335}  (5.43165 × 10^{15,151,335} digits)  
9  387,420,489 (9^{9})  4.28125 × 10^{369,693,099}  (4.08535 × 10^{369,693,099} digits)  
10  10,000,000,000 (10^{10})  10^{10,000,000,000}  (10^{10,000,000,000} + 1 digits) 
Remark: If x does not differ from 10 by orders of magnitude, then for all . For example, in the above table, and the difference is even smaller for the following rows.
Extensions
[edit]Tetration can be extended in two different ways; in the equation , both the base a and the height n can be generalized using the definition and properties of tetration. Although the base and the height can be extended beyond the nonnegative integers to different domains, including , complex functions such as , and heights of infinite n, the more limited properties of tetration reduce the ability to extend tetration.
Extension of domain for bases
[edit]Base zero
[edit]The exponential is not consistently defined. Thus, the tetrations are not clearly defined by the formula given earlier. However, is well defined, and exists:^{[10]}
Thus we could consistently define . This is analogous to defining .
Under this extension, , so the rule from the original definition still holds.
Complex bases
[edit]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.^{[citation needed]}
Extensions of the domain for different heights
[edit]Infinite heights
[edit]Tetration can be extended to infinite heights; 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.^{[11]} 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 exponential of x does not exist when x > e^{1/e} because the maximum of y^{1/y} is e^{1/e}. The limit also fails to exist when 0 < x < e^{−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 on the positive real line, 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
[edit]We can use the recursive rule for tetration,
to prove :
Substituting −1 for k gives
 .^{[12]}
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.^{[12]}
For , any definition of is consistent with the rule because
 for any .
Real heights
[edit]This article includes a list of general references, but it lacks sufficient corresponding inline citations. (July 2019) 
At this time there is no commonly accepted solution to the general problem of extending tetration to the real or complex values of n. 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 ^{[13]}
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
[edit]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^{[6]} states: Let . If is continuous and satisfies the conditions:
 is differentiable on (−1, 0),
 is a nondecreasing or nonincreasing function on (−1, 0),
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 (−1, 0),
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 (−1, +∞). All other sufficientlydifferentiable solutions must have an inflection point on the interval (−1, 0).
Higher order approximations for real heights
[edit]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.^{[1]}
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.^{[1]}^{[14]}
Complex heights
[edit]In 2017, it was proven^{[15]} 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.^{[16]} The construction of such a function was originally demonstrated by Kneser in 1950.^{[17]} 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 greater than . Subsequent work extended the construction to all complex bases.^{[18]}
The requirement of the tetration being holomorphic is important for its uniqueness. Many functions S can be constructed as
where α and β are real sequences which decay fast enough to provide the convergence of the series, at least at moderate values of Im z.
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.
Ordinal tetration
[edit]Tetration can be defined for ordinal numbers via transfinite induction. For all α and all β > 0:
Nonelementary recursiveness
[edit]Tetration (restricted to ) is not an elementary recursive function. One can 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
[edit]Exponentiation has two inverse operations; roots and logarithms. Analogously, the inverses of tetration are often called the superroot, and the superlogarithm (In fact, 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 superlogarithm base y of x.
Superroot
[edit]The superroot is the inverse operation of tetration with respect to the base: if , then y is an nth superroot of x ( or ).
For example,
so 2 is the 4th superroot of 65,536.
Square superroot
[edit]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:^{[19]}
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 does not have any real square superroots, but the formula given above yields countably infinitely many complex ones for any finite x not equal to 1.^{[19]} The function has been used to determine the size of data clusters.^{[20]}
At :
Other superroots
[edit]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.
One of the simpler and faster formulas for a thirddegree superroot is the recursive formula, if: x^{xx} = a, and next x (n + 1) = exp (W (W (x (n) ln (a)))), for example x (0) = 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 superroots can be based on the normal roots: "cube superroots" can be expressed as ; the "4th superroot" can be expressed as ; and the "nth 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 n = ∞, 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.^{[21]} It is still an open question whether irrational superroots are transcendental in the latter case.
Superlogarithm
[edit]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
[edit]Other than the problems with the extensions of tetration, 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 either of ^{4}π or ^{5}e is an integer.^{[22]}^{[additional citation(s) needed]}
 It is not known whether ^{n}q is rational for any positive integer n and positive noninteger rational q.^{[21]} For example, it is not known whether the positive root of the equation ^{4}x = 2 is a rational number.^{[citation needed]}
 It is not known whether ^{e}π or ^{π}e are rationals or not.
Applications
[edit]For each graph H on h vertices and each ε > 0, define
Then each graph G on n vertices with at most n^{h}/D copies of H can be made Hfree by removing at most εn^{2} edges.^{[23]}
See also
[edit] Ackermann function
 Big O notation
 Double exponential function
 Hyperoperation
 Iterated logarithm
 Symmetric levelindex arithmetic
Notes
[edit] ^ Rudolf von Bitter Rucker's (1982) notation ^{n}x, as introduced by Hans Maurer (1901) and Reuben Louis Goodstein (1947) for tetration, must not be confused with Alfred Pringsheim's and Jules Molk's (1907) notation ^{n}f(x) to denote iterated function compositions, nor with David Patterson Ellerman's (1995) ^{n}x presuperscript notation for roots.
References
[edit] ^ ^{a} ^{b} ^{c} ^{d} Neyrinck, Mark. An Investigation of Arithmetic Operations. Retrieved 9 January 2019.
 ^ R. L. Goodstein (1947). "Transfinite ordinals in recursive number theory". Journal of Symbolic Logic. 12 (4): 123–129. doi:10.2307/2266486. JSTOR 2266486. S2CID 1318943.
 ^ N. Bromer (1987). "Superexponentiation". Mathematics Magazine. 60 (3): 169–174. doi:10.1080/0025570X.1987.11977296. 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 0994348.
 ^ Weisstein, Eric W. "Power Tower". MathWorld.
 ^ ^{a} ^{b} Hooshmand, M. H. (2006). "Ultra power and ultra exponential functions". Integral Transforms and Special Functions. 17 (8): 549–558. doi:10.1080/10652460500422247. S2CID 120431576.
 ^ "Power Verb". J Vocabulary. J Software. Retrieved 20111028.
 ^ "Spaces". Retrieved 20220217.
 ^ DiModica, Thomas. Tetration Values. Retrieved 15 October 2023.
 ^ "Climbing the ladder of hyper operators: tetration". math.blogoverflow.com. Stack Exchange Mathematics Blog. Retrieved 20190725.
 ^ 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} Müller, M. "Reihenalgebra: What comes beyond exponentiation?" (PDF). Retrieved 20181212.
 ^ Trappmann, Henryk; Kouznetsov, Dmitrii (20100628). "5+ methods for real analytic tetration". Retrieved 20181205.
 ^ 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".
 ^ Paulsen, W.; Cowgill, S. (March 2017). "Solving in the complex plane" (PDF). Advances in Computational Mathematics. 43: 1–22. doi:10.1007/s1044401795241. S2CID 9402035.
 ^ Kouznetsov, D. (July 2009). "Solution of in complex plane" (PDF). Mathematics of Computation. 78 (267): 1647–1670. doi:10.1090/S0025571809021887.
 ^ Kneser, H. (1950). "Reelle analytische Lösungen der Gleichung und verwandter Funktionalgleichungen". Journal für die reine und angewandte Mathematik (in German). 187: 56–67.
 ^ Paulsen, W. (June 2018). "Tetration for complex bases". Advances in Computational Mathematics. 45: 243–267. doi:10.1007/s1044401896157. S2CID 67866004.
 ^ ^{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. arXiv:1809.07369. doi:10.1007/BF02124750. S2CID 29028411.
 ^ Krishnam, R. (2004), "Efficient SelfOrganization Of Large Wireless Sensor Networks" – Dissertation, BOSTON UNIVERSITY, COLLEGE OF ENGINEERING. pp. 37–40
 ^ ^{a} ^{b} Marshall, Ash J., and Tan, Yiren, "A rational number of the form a^{a} with a irrational", Mathematical Gazette 96, March 2012, pp. 106–109.
 ^ Bischoff, Manon (20240124). "A Wild Claim about the Powers of Pi Creates a Transcendental Mystery". Scientific American. Archived from the original on 20240424. Retrieved 20240423.
 ^ Jacob Fox, A new proof of the graph removal lemma, arXiv preprint (2010). arXiv:1006.1300 [math.CO]
 Daniel Geisler, Tetration
 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. Much information on the Lambert W function, Riemann surfaces, and analytic continuation.)
 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 (4): 235–252. doi:10.1080/00029890.1981.11995239.
 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.)
 The Fourth Operation
 Luca Moroni, The strange properties of the infinite power tower (https://arxiv.org/abs/1908.05559)
Further reading
[edit] Galidakis, Ioannis; Weisstein, Eric Wolfgang. "Power Tower". MathWorld. Retrieved 20190705.