If iterated exponentiation (usually also called "Powertower") is understood as mathematical operator or as function depending on the iteration height, then this operator/operation/function is often called "tetration" and is assumed as next step in the operator hierarchy addition,multiplication, ...

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101
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1answer
20k views

Can $x^{x^{x^x}}$ be a rational number?

If $x$ is a positive rational number, but not an integer, then can $x^{x^{x^x}}$ be a rational number ? We can prove that if $x$ is a positive rational number but not an integer, then $x^x$ can ...
66
votes
4answers
4k views

Are these solutions of $2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$ correct?

Find $x$ in $$ \Large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$$ A trick to solve this is to see that $$\large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}} \quad\implies\quad 2 = ...
43
votes
11answers
4k views

Why are addition and multiplication commutative, but not exponentiation?

We know that the addition and multiplication operators are both commutative, and the exponentiation operator is not. My question is why. As background there are plenty of mathematical schemes that ...
30
votes
1answer
647 views

How do I calculate the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$

I need to find the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$ ( i.e. $\lfloor\{e^{e^{e^{e^{e}}}}\}^{-1}\rfloor$), or even higher towers. The number is too big to ...
27
votes
1answer
415 views

Iterated exponent of $i$

WolframAlpha seems to tell me that $e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^i}}}}}}}}}} = 1$, see link. Is this just an error or is it for real? Adding one more $e$ to the bottom of the tower gives me the ...
25
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2answers
2k views

Is There a Natural Way to Extend Repeated Exponentiation Beyond Integers?

This question has been in my mind since high school. We can get multiplication of natural numbers by repeated addition; equivalently, if we define $f$ recursively by $f(1)=m$ and $f(n+1)=f(n)+m$, ...
25
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1answer
2k views

Infinite tetration, convergence radius

I got this problem from my teacher as a optional challenge. I am open about this being a given problem, however it is not homework. The problem is stated as follows. Assume we have an infinite ...
24
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4answers
2k views

Seems that I just proved $2=4$.

Solving $x^{x^{x^{.^{.^.}}}}=2\Rightarrow x^2=2\Rightarrow x=\sqrt 2$. Solving $x^{x^{x^{.^{.^.}}}}=4\Rightarrow x^4=4\Rightarrow x=\sqrt 2$. Therefore, $\sqrt 2^{\sqrt 2^{\sqrt 2^{.^{.^.}}}}=2$ and ...
22
votes
2answers
787 views

What is the derivative of ${}^xx$

How would one find: $$\frac{\mathrm d}{\mathrm dx}{}^xx?$$ where ${}^ba$ is defined by $${}^ba\stackrel{\mathrm{def}}{=}\underbrace{ a^{a^{\cdot^{\cdot^{\cdot^a}}}}}_{\text{$b$ times}}$$ Work so ...
22
votes
1answer
1k views

$n^{th}$ derivative of a tetration function

I stumbled upon this very peculiar function last summer, namely: $f(x)=x^{x^{x^{...^{x}}}}$, where there is a number $n$ of $x$'s in the exponent, I tried to find the derivative for the function and I ...
21
votes
8answers
735 views

Infinite powering by $i$ [duplicate]

Find the value of: $i^{i^{i^{i^{i^{i^{....\infty}}}}}}$ Simply infinite powering by i's and the limiting value. Thank you for the help.
21
votes
3answers
1k views

Explain $x^{x^{x^{{\cdots}}}} = \,\,3$

$$x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} =\quad 2$$ This equation has the answer $\sqrt{2}$ by taking $\log$ to both side. This answer is correct because I'd proved it by computing the equation repeatedly ...
19
votes
1answer
449 views

Is the positive root of the equation $x^{x^x}=2$, $x=1.47668433…$ a transcendental number?

I can prove using the Gelfond–Schneider theorem that the positive root of the equation $x^{x^x}=2$, $x=1.47668433...$ is an irrational number. Is it possible to prove it is transcendental?
19
votes
3answers
420 views

What combinatorial quantity the tetration of two natural numbers represents?

Tetration is a generalization of exponentiation in arithmetic and a part of a series of other generalized notions, Hyperoperators. Consider $m\uparrow n$ denotes the tetration of $m$ and $n$. i.e. ...
17
votes
4answers
225 views

An obvious pattern to $i\uparrow\uparrow n$ that is eluding us all?

Start with $i=\sqrt{-1}$. This will be $a_1$. $a_2$ will be $i^i$. $a_3$ will be $i^{i^{i}}$. $\vdots$ etc. In Knuth up-arrow notation: $$a_n=i\uparrow\uparrow n$$ And, amazingly, you can ...
16
votes
3answers
375 views

How can I prove $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}…}}}}=2$ [duplicate]

How can I prove $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}...}}}}=2$$ I don't know which method can be used for this?
15
votes
2answers
554 views

What is $i$ exponentiated to itself $i$ times?

I was just wondering about this. I searched about it on the net and found that it is called tetration and after this comes pentation and then hexation and so on so forth. I don't really understand ...
13
votes
2answers
280 views

Proof of strictly increasing nature of $y(x)=x^{x^{x^{\ldots}}}$ on $[1,e^{\frac{1}{e}})$?

The title is fairly self explanatory: I have been trying to rigorously prove that $y(x)=x^{x^{x^{\ldots}}}$ is a strictly increasing function over the interval $[1,e^{\frac{1}{e}})$ for a while now, ...
13
votes
2answers
591 views

A puzzle with powers and tetration mod n

A friend recently asked me if I could solve these three problems: (a) Prove that the sequence $ 1^1, 2^2, 3^3, \dots \pmod{3}$ in other words $\{n^n \pmod{3} \}$ is periodic, and find the length of ...
12
votes
1answer
356 views

Is it possible to prove the positive root of the equation ${^4}x=2$, $x=1.4466014324…$ is irrational?

(somewhat related to my earlier question) Let ${^n}a$ denote tetration $\underbrace{a^{a^{.^{.^{.^a}}}}}_{n \text{ times}}$ (or, defined recursively, ${^1}a=a$, ${^{n+1}}a=a^{({^n}a)}$). The ...
12
votes
2answers
625 views

Derivative of $x^{x^{\cdot^{\cdot}}}$?

The infinite tetration is defined as $$f(x)=x^{x^{\cdot^{\cdot}}}$$ This function is defined for $e^{-e} \leq x \leq e^{e-1}$. (Wikipedia image) Can one determine the derivative of this function? ...
12
votes
3answers
2k views

How to evaluate fractional tetrations?

Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the ...
12
votes
9answers
2k views

half iterate of $x^2+c$

I'm looking for literature on fractional iterates of $x^2+c$, where c>0. For c=0, generating the half iterate is trivial. $$h(h(x))=x^2$$ $$h(x)=x^{\sqrt{2}}$$ The question is, for $c>0,$ and ...
11
votes
4answers
472 views

Solve $x^{x^x}=-1$

I do know that if you use tetration the equation would look like this. $$^3x=-1$$ You could then theoretically use the super-root function to solve the equation, but I do not know how to use the ...
11
votes
2answers
627 views

Complex towers: $i^{i^{i^{…}}}$

If $w = z^{z^{z^{...}}}$ converges, we can determine its value by solving $w = z^{w}$, which leads to $w = -W(-\log z))/\log z$. To be specific here, let's use $u^v = \exp(v \log u)$ for complex $u$ ...
11
votes
1answer
162 views

Prime factors of $\sum_{k=1}^{30}k^{k^k}$

I checked the prime factors of $$\sum_{k=1}^{30}k^{k^k}$$ and did not find any upto $10^8$ Are there any useful restrictions to accelerate the search ?
10
votes
4answers
344 views

If $y=x^{x^{x^{x^{x^{.^{.^{.}}}}}}}$ then how $y=x^y$?

In questions like, find the derivative of $f(x)=x^{x^{x^{x^{x^{.^{.^{.}}}}}}}$, how can we formally show that $y=x^y$? We use this technique for all type of iterations, e.g. ...
10
votes
2answers
614 views

Example of Tetration in Natural Phenomena

Tetration is a natural extension of the concept of addition, multiplication, and exponentiation. It is quite obvious that there are things in the physics world which can be modeled by these 3 lowest ...
9
votes
3answers
754 views

How to calculate $f(x)$ in $f(f(x)) = e^x$?

How would I calculate the power series of $f(x)$ if $f(f(x)) = e^x$? Is there a faster-converging method than power series for fractional iteration/functional square roots?
9
votes
1answer
146 views

Prove that $F_{2n}(x)=0$ has exactly one root in the interval $x\in(0,1),$ and this root $\to 0$ when $n \to \infty.$

Define $$f_1(x)=x\\f_2(x)=x^x\\\vdots\\f_{n+1}(x)=x^{f_n(x)}$$ Let $F_n(x)=f_n^{'}(x).$ Hence $$F_1(x)=1\\F_2(x)=x^x(1+\log(x))\\\vdots$$ Prove that $F_{2n}(x)=0$ has exactly one root in the ...
8
votes
3answers
2k views

Calculating 7^7^7^7^7^7^7 mod 100

What is $$\large 7^{7^{7^{7^{7^{7^7}}}}} \pmod{100}$$ I'm not much of a number theorist and I saw this mentioned on the internet somewhere. Should be doable by hand.
8
votes
1answer
286 views

A problem in understanding infinite towers (tetration)

To solve equations involving power towers (infinite tetration) we usually do something like this: $$x^{x^{x^{x^{\dots}}}} =k$$ $$x^{(x^{x^{x^{\dots}}})} =k$$ $$x^k=k$$ $$x=\sqrt[k]k$$ But what if ...
8
votes
0answers
212 views

Largest $x$ such that the power tower (tetration) $x^{x^{x^{x^{…}}}}$ converges? [duplicate]

Possible Duplicate: Infinite tetration, convergence radius Recently in this thread, Pseudo Proofs that are intuitively reasonable, I learned that ...
7
votes
2answers
455 views

Continuum between addition, multiplication and exponentiation?

I noticed this old post which attempts to find the shades of grey between a linear and log scale where results are between zero and one. However, I was looking for the more general case where we find ...
7
votes
2answers
175 views

Do the last digits of exponential towers really converge to a fixed sequence?

While fooling around with exponential towers I noticed something odd: $$ 3^{3} \equiv 2\underline{7} \mod 100000 $$ $$ 3^{3^{3}} \equiv 849\underline{87} \mod 100000 $$ $$ 3^{3^{3^{3}}} \equiv ...
7
votes
0answers
91 views

How to solve $x^x = y^y \mod p$?

Let $p>5$ be a prime. How to solve $x^x = y^y \mod p$? How many solutions are there for a given $p$ such that $x,y < p$? I know the discrete logarithm and the theory of quadratic residues, but ...
7
votes
0answers
276 views

Prime factor of $2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4$

I checked the prime factors of $$2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4 = 2^{2^{2^2}} + 3^{3^{3^3}}$$ with trial division and found non below $8*10^9$ Nevertheless, the given number has still ...
6
votes
2answers
558 views

Uniqueness of Tetration

Let $f(0)=1$ and $f(x+1)=2^{f(x)}$ Also let f be infinitely differentiable. Then does f exist and is it unique? If f is merely continuous, then any continuous function such that f(0)=1 f(1)=2 ...
6
votes
1answer
574 views

Solutions of $f(f(z)) = e^z$

It is my impression that if we find a function f(z) that satisfies $$f(f(z)) = e^z $$ there is only one point z that satisfies the relation. This dawned on me when I noticed that the pesky z that ...
6
votes
1answer
136 views

Growth of $n!!\dots !$

The asymptotic growth of the factorial function $n!$ is famously given by Stirling's formula as $$n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n$$ Is there a similar formula for the iterated ...
6
votes
4answers
225 views

A limit related to super-root (tetration inverse).

Recall that tetration ${^n}x$ for $n\in\mathbb N$ is defined recursively: ${^1}x=x,\,{^{n+1}}x=x^{({^n}x)}$. Its inverse function with respect to $x$ is called super-root and denoted $\sqrt[n]y_s$ ...
6
votes
3answers
431 views

Algorithm for tetration to work with floating point numbers

So far, I've figured out an algorithm for tetration that works. However, although the variable a can be floating or integer, unfortunately, the variable ...
6
votes
0answers
255 views

Notation for n-ary exponentiation

We have $n$-ary sums ($\sum$) and products ($\prod$). Is there an $n$-ary exponentiation operator? $$\underset{i=1}{\overset{n}{\LARGE{\text{E}}}}\, x_i = x_1 \text{^} (x_2 \text{^} (\cdots \text{^} ...
6
votes
0answers
335 views

Is ${^5\pi}$ an integer? [duplicate]

Possible Duplicate: How to show $e^{e^{e^{79}}}$ is not an integer Is ${^5\pi}$ an integer? It is "obviously" not, right? But can we prove it? Here ${^5\pi}$ means the result of tetration ...
5
votes
2answers
170 views

Last digits of power towers $7$, $7^7$, $7^{7^7}$, $7^{7^{7^7}}$, … don't change, and generalisation

While playing around with Wolfram Alpha, I noticed that the last four digits of $7^{7^{7^{7^7}}}, 7^{7^{7^{7^{7^7}}}},$ and $7^{7^{7^{7^{7^{7^7}}}}}$were all $2343$. In fact, the number of sevens did ...
5
votes
2answers
671 views

Last few digits of $n^{n^{n^{\cdot^{\cdot^{\cdot^n}}}}}$

I want to compute last few digts (as much as possible ) of the following number $$ N:=n^{n^{n^{\cdot^{\cdot^{\cdot^n}}}}}\!\!\!\hspace{5 mm}\mbox{ if there are $k$ many $n$'s in the expression and ...
5
votes
2answers
69 views

Checking primality for $2 \uparrow \uparrow n + 3 \uparrow \uparrow n$

Is there a clever way to test primality for $$2 \uparrow \uparrow n \quad + \quad 3 \uparrow \uparrow n$$ where $n \gt 3$? (Not surprisingly, I got stuck after that) For $n \le 3$ we get: $$ ...
5
votes
2answers
246 views

radius of convergence of half iterate of sinh(z)?

The half iterate of sinh(z) has a formal power series, centered around z=0. Does the formal power series for the half iterate converge at the origin? This is equivalent to asking if the half iterate ...
5
votes
1answer
152 views

Solving for $a$ in power tower equation

$$n=a^{(a+1)^{(a+2)^{(a+3)\cdots}}}$$ How would one go about solving in this equation? I am more used to solving equations in this form: $$n=a^{a^{a^{a\cdots}}}$$ Which you solve in this form: ...
5
votes
2answers
54 views

Finding the Final Digits in a Stacked Exponents Problem

How to find the the last two digits of the number $$\underbrace{\huge 7^{7^{7^{...}}}}_{1+n}$$ When there are $n$ sevens in the ascending exponents. I have no idea how to condense this or ...