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.

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9
votes
2answers
418 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\...
2
votes
1answer
62 views

Maximize $a_1^{a_2^{\ldots^{a_n}}}$, where $(a_1,a_2,\ldots,a_n)$ is a permutation of $(b_1,b_2,\ldots,b_n)$

You are given a tuple of integers $B=(b_1,b_2,\ldots,b_n)$. Find $(a_1,a_2,\ldots,a_n)$ - a permutation of $(b_1,b_2,\ldots,b_n)$ - that maximizes $a_1^{a_2^{\ldots^{a_n}}}$. For example - If $B=...
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 ...
8
votes
4answers
465 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 $...
10
votes
3answers
463 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....
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 $2^{2 \cdot 5^6}+3^{5^4} \cdot 2^{5^4}= 2^{5^4} \cdot ( 2^{2 \cdot 5^6 -5^4}+ 3^{5^4} )$.
0
votes
1answer
50 views

Compute the product of digits of P

Give $P$ a integer number where $$P=2^{3^{4^{5^{\dots1000}}}}$$ Then Compute The product of dígits of $P$ Compute $P\pmod{5}$ for The segond i think its will be something like $$\...
0
votes
1answer
66 views

if -a^(-b^-c) is a positive integer and a, b, and c are integers, then…

(a) a must be negative (b) b must be negative (c) c must be negative (d) b must be an even positive integer (e) none of the above
11
votes
3answers
5k 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.
1
vote
2answers
225 views

calculate value of $2^{3^{4^{5}}}/e^{10240}$

I am trying to calculate the value of $b=\dfrac{2^{3^{4^5}}}{e^{10240}}$. Is there any method to solve this efficiently?
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 ...
1
vote
1answer
88 views

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 $g=\...
0
votes
4answers
55 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.
15
votes
2answers
898 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, $$s_n=\underbrace{\sqrt2^{\sqrt2^{\ .^{\ .^{\ .^{\sqrt2}}}}}}_{n\ \text{levels}}.$$ ...
7
votes
2answers
274 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 ...
5
votes
0answers
780 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 ...
0
votes
1answer
53 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 ...
1
vote
1answer
799 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 ...
4
votes
2answers
8k views

x raised to itself infinite number of times [duplicate]

$$\Large x^{x^{x^{x^{x^{.^{\,.^{\,.}}}}}}} = 2$$ What is $x$?
2
votes
1answer
205 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) ...
1
vote
0answers
23 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 ...
1
vote
0answers
53 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 ...
2
votes
0answers
159 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 ...
7
votes
0answers
286 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 ...
16
votes
2answers
679 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^...
2
votes
1answer
226 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 ...
4
votes
0answers
140 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 $f_4(x)=x^{(x^{(x^x)})}$....
23
votes
1answer
575 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). ...
4
votes
1answer
210 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 ...
91
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}}}}} < ...
7
votes
1answer
946 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: $$2 \uparrow\uparrow 10 \pmod{10^9}$$
2
votes
1answer
566 views

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

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 ...
14
votes
2answers
542 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 ...