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

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Transforming a power tower to a product

It is possible to write the product of a sequence of terms $a_i$ as a function of the sum of a sequence of functions of these terms: $$\prod_i a_i=f\left(\sum_i g(a_i)\right)$$ where $f=\exp$ and ...
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4answers
48 views

How to multiply two different numbers with different powers

How do you multiply and simplify: $\left(\frac{2}{3}\right)^{1/6}\cdot 18^{1/3}$? Simplify in surd form.
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2answers
161 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, $$s_n=\underbrace{\sqrt2^{\sqrt2^{\ .^{\ .^{\ .^{\sqrt2}}}}}}_{n\ \text{levels}}.$$ ...
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0answers
13 views

Solving complicated double exponential relations

I have a table of values representing what I suspect to be a continuous function that looks like: $$f(x) = 2^{2^{ax+b}} + 2^{(-2)^{cx+d}} + k$$ I'm not entirely certain that this is correct, but I ...
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1answer
20 views

Precedence or powers - Is there more than one way

I'm developing a calculator and I've encountered this issue: 2222 which power should be calculated first, is there a way this could be solved from the left to the right, meaning 422 and so on.... I ...
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1answer
141 views

Double exponential Taylor series $\exp(-\exp(k-ex))$

k is real constant $\gt = 1$. Is $a_n$ for $f(x)$ positive, increasing, and $\lt 1$, where $n\lt= e^{k-1}$? $$f(x) = \sum_{n=0}^{\infty} a_n x^n = \exp(-\exp(k-ex))$$ $f(x)$ is the double ...
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0answers
247 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 ...
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2answers
172 views

x raised to itself infinite number of times [duplicate]

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

Find the last two digits of 9^(9^9) [duplicate]

I want to find the last two digits of $9^{9^9}$, that is $9$ raised to the power $9^9$. I tried using Euler's theorem but I can't make anything of it. As always, I ask only for a minor hint, not a ...
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0answers
17 views

Sufficient condition for an equivalence

What is a sufficient condition for the equivalence $$ a_1 \uparrow a_2 \uparrow ... \uparrow a_n \equiv a_1 \uparrow a_2 \uparrow ... \uparrow a_n \uparrow a_{n+1}\ mod(\ m)\ ? $$ In a closely ...
2
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1answer
118 views

Prime factors of a sum of special power towers

Denote $$x(n) = 2 \uparrow 3 \uparrow ... \uparrow n$$ and $$y(n) = n \uparrow (n-1) \uparrow ... \uparrow 3 \uparrow 2$$ Finally denote $$z(n) = x(n) + y(n)$$ So, the first few numbers z(n) ...
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0answers
47 views

Summation of $2^{(-2^{n})}$ [duplicate]

By the ratio test, I know that this series convernges: $\sum2^{(-2^{n})}$, in the limit $n$ goes to infinity. Probably to something close to $.8$ (if not equal to $.8$). The problem is, how do I ...
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2answers
465 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 ...
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0answers
72 views

Special power towers

Let $x(n)$ be the power tower $2 \uparrow 3 \uparrow 4 \uparrow \cdots \uparrow n$. Let $y(n)$ be the power tower $n \uparrow (n-1) \uparrow \cdots \uparrow 3 \uparrow 2$ My questions : Is there a ...
2
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1answer
185 views

Inverse function of $x\mapsto \sqrt[x]x$ on $\left[0,e^{-1}\right]$

Why is it, that the inverse of $\sqrt[x]x$ is given by the infinite power tower in $x\in[\frac1e;e]$, but not in $x\in[0;\frac1e]$? I know that the power tower diverges on that interval, but even if ...
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0answers
115 views

On the sequence $(f_n)$ defined by $f_1(x)=x$ and $f_{n+1}(x)=x^{f_n(x)}$

Consider the numbers $x^x$,$x^{(x^x)}$,$x^{(x^{(x^x)})}$, etc. Let $n$ be the number of times $x$ appears in the power tower and $f_n$ the corresponding function, for example ...
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1answer
307 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). ...
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1answer
534 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 ...
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1answer
125 views

Question involving exponential tower of 19

Consider: $$ y = \underbrace{19^{19^{\cdot^{\cdot^{\cdot^{19}}}}}}_{101 \text{ times}} $$ with the tower containing a hundred $ 19$s. Take the sum of the digits of the resulting number. Again, add the ...
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0answers
112 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 $f_1(x)=x,f_2(x)=x^x,\dots f_{n+1}(x)=x^{f_n(x)}~(n\geq 1).$ Let $F_n(x)=f_n^{'}(x).$ Hence $F_1(x)=1, F_2(x)=x^x(1+\log(x))\dots.$ Prove that $F_{2n}(x)=0$ has exactly one root in the ...
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1answer
145 views

Fixed Point of $x_{n+1}=i^{x_n}$

For $x \in \Bbb C$, let $f(x)=i^x = \exp(i\pi x)$, where $i^2=-1$. Then find the fixed points for $f$. EDIT: Let for all $n\geq 1$ $$\large a_n=\underbrace{i^{i^{\cdots i}}}_{\text{$n$ times}}$$ My ...
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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 ...
5
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3answers
750 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.
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2answers
532 views

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

how to integrate $$\int\underbrace{x^{x^{\cdot^{\cdot^x}}}}_ndx$$ $\color{red}{\text{or how to calculate this integral when its bounded}}$ ...
13
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2answers
301 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 ...
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0answers
190 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 ...
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2answers
137 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 ...
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1answer
374 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: $$2 \uparrow\uparrow 10 \pmod{10^9}$$
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4answers
2k 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}}}}} < ...