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Questions tagged [power-towers]

For questions pertaining to power towers: expressions like a^(b^(c^d))), which result from iterated exponentiation. The "hyperoperation" tag may be appropriate, too.

115
votes
1answer
21k 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 not ...
90
votes
4answers
4k views

Complexity class of comparison of power towers

Consider the following decision problem: given two lists of positive integers $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_m$ the task is to decide if $a_1^{a_2^{\cdot^{\cdot^{\cdot^{a_n}}}}} < ...
75
votes
4answers
7k 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 = x^{\Big(x^{x^{x^{...
50
votes
1answer
1k views

How many values of $2^{2^{2^{.^{.^{.^{2}}}}}}$ depending on parenthesis?

Suppose we have a power tower consisting of $2$ occurring $n$ times: $$\huge2^{2^{2^{.^{.^{.^{2}}}}}}$$ How many values can we generate by placing any number of parenthesis? It is fairly simple for ...
37
votes
4answers
2k views

Which is bigger: $9^{9^{9^{9^{9^{9^{9^{9^{9^{9}}}}}}}}}$ or $9!!!!!!!!!$?

In my classes I sometimes have a contest concerning who can write the largest number in ten symbols. It almost never comes up, but I'm torn between two "best" answers: a stack of ten 9's (exponents) ...
37
votes
2answers
1k views

A new interesting pattern to $i↑↑n$ that looks cool (and $z↑↑x$ for $z\in\mathbb C,x\in\mathbb R$)

Many of you may recall "An obvious pattern to $i↑↑n$ that is eluding us all?", an old question of mine, and just recently, I saw this new question that poses a simple extension to tetration at non-...
34
votes
6answers
1k views

What is the derivative of: $f(x)=x^{2x^{3x^{4x^{5x^{6x^{7x^{.{^{.^{.}}}}}}}}}}$?

I happened to ponder about the differentiation of the following function: $$f(x)=x^{2x^{3x^{4x^{5x^{6x^{7x^{.{^{.^{.}}}}}}}}}}$$ Now, while I do know how to manipulate power towers to a certain extent,...
33
votes
4answers
1k 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 ...
27
votes
4answers
1k views

Prove that $2^{2^{\sqrt3}}>10$

With a computer or calculator, it is easy to show that $$ 2^{2^\sqrt{3}} = 10.000478 \ldots > 10. $$ How can we prove that $2^{2^{\sqrt3}}>10$ without a calculator?
27
votes
2answers
1k 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 far ...
25
votes
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 ...
23
votes
1answer
573 views

Convergence of $a_n=(1/2)^{(1/3)^{…^{(1/n)}}}$

The sequence $a_n=(1/2)^{(1/3)^{...^{(1/n)}}}$ doesn't converge, but instead has two limits, for $a_{2n}$ and one for $a_{2n+1}$ (calculated by computer - they fluctuate by about 0.3 at around 0.67). ...
20
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 ...
16
votes
2answers
677 views

how to integrate $\int\underbrace{x^{x^{\cdot^{\cdot^x}}}}_ndx$

how to integrate $$\int\underbrace{x^{x^{\cdot^{\cdot^x}}}}_ndx$$ $\color{red}{\text{or how to calculate this integral when its bounded}}$ $$\color{red}{\int_0^1\underbrace{x^{x^{\cdot^{\cdot^...
15
votes
1answer
4k views

What is wrong with this funny proof that 2 = 4 using infinite exponentiation?

Out of boredom, I decided to recall the following equation: $$x^{x^{x\cdots}} = 2.$$ Which, I simply rewrote like this: $x^2 = 2$, and therefore $x = \sqrt{2}$. Then I took a look at the more ...
14
votes
2answers
539 views

Limit involving power tower: $\lim\limits_{n\to\infty} \frac{n+1}n^{\frac n{n-1}^\cdots}$

What is the value of the following limit? $$\large \lim_{n \to \infty} \left(\frac{n+1}{n}\right)^{\frac{n}{n-1}^{\frac{n-1}{n-2}^{...}}}$$ In general what do limits of infinite decreasing numbers ...
14
votes
2answers
876 views

Convergence of power towers

Let's define the sequence $\{s_n\}$ recursively as $$s_1=\sqrt2,\ \ \ s_{n+1}=\sqrt2^{\,s_n}.$$ Or, in other words, $$s_n=\underbrace{\sqrt2^{\sqrt2^{\ .^{\ .^{\ .^{\sqrt2}}}}}}_{n\ \text{levels}}.$$ ...
11
votes
3answers
4k 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.
11
votes
1answer
162 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 ...
10
votes
2answers
309 views

If $x^{x^4}=4$, then what is the value of $x^{x^2}+x^{x^8}$?

If $x^{x^4}=4$, then what is the value of $x^{x^2}+x^{x^8}$? I tried to simplify it using exponentiation and logs, and even just algebraic manipulation..But I don't know how to do this.
10
votes
3answers
451 views

Number Theory : What are the last three digits of $9^{9^{9^9}}?$

I was doing some basic Number Theory problems and came across this problem and was all thumbs : Find the last three digits of $9^{9^{9^9}}$ How would I go about solving this problem? I am a newbie....
10
votes
3answers
496 views

What is the derivative of $x!^{x!^{x!^{x!^{x!^{x!^{x!^{.{^{.^{.}}}}}}}}}}$

What is the derivative of $$x!^{x!^{x!^{x!^{x!^{x!^{x!^{.{^{.^{.}}}}}}}}}}$$ My effort: Let $$g(x)=x!^{x!^{x!^{x!^{x!^{x!^{x!^{.{^{.^{.}}}}}}}}}}\implies g(x)=x!^{g(x)}$$ Taking natrual log on both ...
9
votes
3answers
359 views

Solutions of $a^{a^x}=x$ for fixed $a>0$

I am interested in the equation $a^{a^x}=x$ for some fixed $a>0$. Is there some way to rearrange for $x$ or solve otherwise? What about the nature of the solutions? For which fixed $a>0$ are ...
9
votes
1answer
469 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 ...
9
votes
2answers
396 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 39\...
8
votes
4answers
453 views

computing ${{27^{27}}^{27}}^{27}\pmod {10}$

I'm trying to compute the most right digit of ${{27^{27}}^{27}}^{27}$. I need to compute ${{27^{27}}^{27}}^{27}(\bmod 10)$. I now that ${{(27)^{27}}^{27}}^{27}(\bmod 10) \equiv{{(7)^{27}}^{27}}^{27} ...
7
votes
2answers
1k 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 $...
7
votes
1answer
206 views

Power towers of $2$ and $3$ - looking for a proof

Let $\uparrow$ denote the right-associative exponentiation operator: $a\uparrow b\uparrow c=a\uparrow(b\uparrow c)=a^{b^c}$ There is a sequence $A248907$ recently submitted to OEIS (see also $A256179$...
7
votes
3answers
199 views

Power Towers, and Notation for Iterated Exponentiation

So far, we use the symbol $$\sum$$ to denote sums, and $$\prod$$ to denote products. But is there any such notation for exponentiation? Has any research been done about exponentiation of this type, ...
7
votes
1answer
938 views

Calculating the residue of power towers

I want to calculate the residue of a power tower. How do I do that? For example, I want to know the answer to this: $$2 \uparrow\uparrow 10 \pmod{10^9}$$
7
votes
2answers
270 views

Power tower inequality

I want to prove the following power tower inequality: $$ 3 \uparrow \uparrow 100 > 4 \uparrow \uparrow 99 $$ but I don't know how to do this. I think that induction will not work, because I think ...
7
votes
1answer
119 views

Attempt on fractional tetration

To start off, I was looking at the following ingeniously made form of the Gamma function: $$\Gamma(z+1)=\lim_{n\to\infty}\frac{n!(n+1)^z}{(1+z)(2+z)\cdots(n+z)}$$ which lies on the back of $$1=\...
7
votes
0answers
283 views

Close power towers

In another question I gave some ways to determine which of two power towers is larger, but my answer there is incomplete because it doesn't handle the case where two towers are very close at each ...
6
votes
4answers
174 views

How to calculate $p={{{{{{6^6}^6}^6}^6}^6}^6}$ $\mod7$?

I've tried two approaches: Approach 1 Since $6 \equiv -1 \pmod7$ So, $p=(-1)^t$ and $t$ is even Therefore, $p=1$. Approach 2 Since $6 \equiv -1 \pmod7$ So, $6^6 \equiv 1 \pmod7$. Hence, ...
6
votes
3answers
304 views

Why is $(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}} \approx\pi$ (matches upto 4 digits).

Is there something deeper here (like this is the truncation of an exact formula or something else going on)?
6
votes
3answers
425 views

Why do $x^{x^{x^{\dots}}}=2$ and $x^{x^{x^{\dots}}}=4$ have the same positive root $\sqrt 2$?

What is $x$ when is satisfies $x^{x^{x^{\dots}}}=2$ ? I am really confused with this; the root is $\sqrt{2}$, but why does the equation $x^{x^{x^{\dots}}}=4$ have the same root?
6
votes
2answers
201 views

Shortcut to $x\uparrow \uparrow n$ for very large $n$ and $x\approx e^{(e^{-1})}$?

If the number $x$ is very close to $e^{(e^{-1})}$ , but a bit larger, for example $x=e^{(e^{-1})}+10^{-20}$, then tetrating $x$ many times can still be small. With $x=e^{(e^{-1})}+10^{-20}$ , even $x\...
6
votes
1answer
95 views

Taylor series of a power tower

I recently proved that the Taylor Series of $\exp(\exp(x))$ is given by $$\exp(\exp(x))=\sum_{n=0}^\infty \frac{eB_n x^n}{n!}$$ where $B_n$ are the Bell Numbers. However, I can't figure out a Taylor ...
6
votes
0answers
77 views

Modular power tower mapping, is it injective?

Given an infinite sequence $a_1, a_2, \dots$ where all $a_i > 1$ we study $a_1^{\,a_2^{\,\cdots}} \bmod m$. While this is an infinite power tower that grows without bound, I argue that it can be ...
5
votes
2answers
511 views

What is the maximum convergent $x$ in the power tower $x^{x^{x^{x\cdots}}}$?

In the power tower $x^{x^{x^{x\cdots}}}$ where there is an infinite stack of $x$'s, what is the maximum convergent number? I know the answer by playing with the form $x^y=y$ and using Mathematica, but ...
5
votes
2answers
167 views

How fast do iterated exponentiation converge?

Iterated exponentiation is defined by $$x \mapsto x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}$$ or more conveniently, we denote by $^rx$ the expression $\underbrace{x^{x^{\cdot^{\cdot^{\cdot^{x}}}}}}_{r \text{ ...
5
votes
1answer
170 views

What is the value of $i^{i^{i^\ldots}}$? [closed]

What is the value of $i^{i^{i^\ldots}}$? My effort is the following: If $z, \alpha \in \mathbb{C}$ with $z \neq 0$ then we can write $z^{\alpha}=e^{\alpha \log z} = e^{\alpha [ \log |z|+i \text{ Arg ...
5
votes
1answer
306 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: $$a^...
5
votes
1answer
75 views

How find the number of zeros at the end of the sum $4^{5^6}+6^{5^4}$?

The problem is to find the number of zeros at the end of the sum $4^{5^6}+6^{5^4}$. I tried $2^{2 \cdot 5^6}+3^{5^4} \cdot 2^{5^4}= 2^{5^4} \cdot ( 2^{2 \cdot 5^6 -5^4}+ 3^{5^4} )$.
5
votes
0answers
57 views

Supremum of arc lengths of graphs of power towers

Consider the set of all functions of one variable $x\in[0,1]$ that can be constructed from any number of instances of that variable using parentheses and exponentiation only: $$x,\;x^x,\,x^{x^x},\;\...
5
votes
0answers
150 views

Peter Walker's $C^{\infty}$ conjectured nowhere analytic slog

Consider the function $h(x)$ which is conjectured to be $c^\infty$ nowhere analytic, and can be used to generate what we call the base change slog, which is the inverse of the basechange sexp. This ...
5
votes
0answers
763 views

Show that the iterated $\ln^{[n]}$ of tetration(x,n) is nowhere analytic

$$f(x) = \lim_{n\to \infty} \ln^{[n]} x \uparrow\uparrow n$$ The conjecture is that $f(x)$ is monotonic and infinitely differentiable at the real axis, but nowhere analytic; because at each point on ...
4
votes
2answers
8k views

x raised to itself infinite number of times [duplicate]

$$\Large x^{x^{x^{x^{x^{.^{\,.^{\,.}}}}}}} = 2$$ What is $x$?
4
votes
3answers
182 views

The smallest number $m$, such that $m\uparrow \uparrow (n+1)>n\uparrow\uparrow n$

A natural number $n\ge 3$ is given. Denote $a\uparrow\uparrow b$ to be a power tower of $b$ $a's$. Let $m$ be the smallest natural number , such that $m\uparrow\uparrow(n+1) > n\uparrow\uparrow n$ ...
4
votes
2answers
131 views

Magnitude of $f_3(n)$ compared to power towers of tens

In the fast growing hierarchy , the sequence $f_2(n)$ is defined as $$f_2(n)=n\cdot 2^n$$ The number $f_3(n)$ is defined by $$f_3(n)=f_2^{\ n}(n)$$ For example, to calculate $f_3(5)$, we have to ...