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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3
votes
3answers
84 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 $m\uparrow\uparrow(n+1) > n\uparrow\uparrow n$ ...
3
votes
1answer
54 views

Solving $x^{1/x}=y^{1/y}$ iteratively by power towers, convergence

Suppose we have no idea that $x^{1/x}=\exp \left( \frac{1}{x} \ln x \right)$ or don't know anything about exponents and logarithms. But we can certainly compute roots, for example by this method. ...
3
votes
2answers
67 views

How fast does this sequence grow?

I have the following recursive definition of a sequence of numbers: $$a_{n+1}=(a_n)^{(a_{n-1})}$$ And $a_0=a_1=2$. The first few terms are: $$a_2=4$$ $$a_3=16$$ $$a_4=65536$$ $$a_5=1.1579209 ...
1
vote
0answers
38 views

Convergence of power tower $(1/2)^{(1/3)^{\dots (1/n)}}$

I was wondering whether it is possible to evaluate the following limit $\lim_{n \to \infty} a_n = \lim_{n \to \infty} ...
0
votes
3answers
214 views

Differentiating $4^{x^{x^x}}$

$4^{x^{x^x}}$ Hi, I came across this question and would like to check whether I have it done correctly: $e^{x^3}\ln4=4^{x^3}(3\ln4\cdot x^2)$ is this the correct solution?
2
votes
2answers
217 views

Integrate $x$ to the power $x$… to the power $x$… infinitely

This came across my mind, integrating $x$ to the power $x$ infinitely, I couldn't find anything on it. $$\Large \int x^{x^{x^{x\,\cdots}}} \, dx$$ How would you go about this?
0
votes
0answers
25 views

What is the generating function $G(x,y)$ for powertowers of $x$?

I'm looking for a generating function $G(x,y)$ for powertowers of $x$ such that $G(x,y) = 1 + xy + x^x {y^2 \over 2!} + x^{x^x} {y^3 \over 3!} + x^{x^{x^x}} {y^4 \over 4!} + ...$ Is there any ...
0
votes
0answers
29 views

What the minimum of infinite tetration divided by $\sqrt{x}$?

For some values of $x$ the limit of infinite tetration converges. For example when $x=\sqrt{2}$ this is fixed point $$\lim_{n\rightarrow \infty} \sqrt{2} \uparrow \uparrow n = ...
19
votes
4answers
480 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 ...
2
votes
1answer
58 views

Any way we can evaluate the infinite power tower where it diverges?

When you have: $$x=y^{y^{y^{y\dots}}}$$ You have: $$x=e^{-W(-\ln(y))}$$ ONLY when the power tower converges. But what about when it doesn't? Is there any way to justify ...
2
votes
0answers
41 views

Does $y=\lim_{n\to\infty}x\uparrow\uparrow n-f(x,n)$ have a root at $2$?

While messing around on Desmos calculator, I found an interesting factorial/tetration function, where we are using Nutch arrow notation. $$y=\lim_{n\to\infty}x\uparrow\uparrow ...
4
votes
1answer
46 views

Extra root of hyperpower equations [duplicate]

Consider the hyperpower equation $$x^{x^{x^{x...}}}=2$$ We will use the method that Let $y=x^{x^{x^{x...}}}$ so $x^y=x^{x^{x^{x...}}}=2$ and $x^2=2$ to give the solution $x=\sqrt2$ However ...
32
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 ...
9
votes
2answers
293 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 $$g(x)=x!^{x!^{x!^{x!^{x!^{x!^{x!^{.{^{.^{.}}}}}}}}}}=>g(x)=x!^{g(x)}$$ Taking natrual log on both ...
1
vote
2answers
159 views

Could you solve $x↑^n2=x↑^m2$?

As my title asks, could you solve $x^2=2^x$? But that's the worrisome part, as I noticed $x↑^n 2=x↑^m 2$ and $2↑^p x=2↑^q x$ will always have a solution at $x=2$. However, there is bound to be at ...
9
votes
2answers
158 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.
1
vote
0answers
74 views

Solving for $x$ in $M ^ {M ^ M} = x ^ {1 / (x-1)}$ where $M = 5 ^ {\sqrt{5} / 10}$

Methods used by analogy, for example $x ^ x = 3 ^ 3 \implies x = 3$, Determine the value of $x$ in $$M ^ {M ^ M} = x ^ {1 / (x-1)}$$ if $M = 5 ^ {\sqrt{5} / 10}$.
5
votes
1answer
162 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: ...
0
votes
0answers
20 views

Check whether a number could expressed as power of another two numbers [duplicate]

I found in many places how to find whther a number could be expressed as power of 2. What I need to know is, if a number is given whther that number could be expressed as a number raised to another. ...
-2
votes
1answer
115 views

Does there exist a prime that is a sum of two prime power towers? [closed]

Does there exist prime number of the form $$\huge 2^{3^{5^{\,.^{.^{.\,^{p_n}}}}}} + p_n^{p_{n-1}^{\,.^{.^{.\,^{3^{2}}}}}}$$ where $p_n$ is the $n$-th prime number(and both towers are running through ...
9
votes
1answer
309 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 ...
0
votes
0answers
67 views

Inverse and named fixed values, with ↑↑?

The inverse of $+$ is $-$, of $\times$ is $/$ and of $\text{^}$ is Log. Continuing upwards hyperoperationally, what is the inverse of $↑↑$? Whats more somtimes values that are fixed are given ...
2
votes
1answer
129 views

Remainder of a power tower under modulo $2013$

I have an expression like this: $$\left(\large 6000^{5999^{5998^{5997^{{\ldots^{1}}}}}}\right)\bmod 2013$$ Then which method should I use to solve it? Please provide the method not the answer. ...
7
votes
1answer
154 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 ...
2
votes
1answer
48 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 ...
7
votes
2answers
190 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 ...
10
votes
3answers
250 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 ...
5
votes
1answer
71 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 $2^{2 \cdot 5^6}+3^{5^4} \cdot 2^{5^4}= 2^{5^4} \cdot ( 2^{2 \cdot 5^6 -5^4}+ 3^{5^4} )$.
25
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?
0
votes
1answer
44 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
56 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
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 ...
3
votes
0answers
46 views

Iterations $n, n^n, (n^n)^{(n^n)},…$

(Note: I'm reposting this, as I posted the original too late in the evening to gain anyone's notice.) A contest problem (#2 on the 2010 Virginia Tech Math Competition) proffers the solver the ...
1
vote
2answers
172 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?
1
vote
1answer
64 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 ...
0
votes
4answers
54 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.
14
votes
2answers
364 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, $$s_n=\underbrace{\sqrt2^{\sqrt2^{\ .^{\ .^{\ .^{\sqrt2}}}}}}_{n\ \text{levels}}.$$ ...
0
votes
1answer
32 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
377 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 ...
0
votes
0answers
101 views

Condition for existence of solution to a power-tower equation.

"For two positive number a, b which satisfies the condition $\ln a \ln b <0$. equation $a^{b^{a^{b^x}}}=x$ has only one root if and only if ${\frac{d}{dx}a^{b^x}}_{x=t}\geq-1$, where $a^{b^t}=t$" ...
4
votes
0answers
471 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 ...
2
votes
2answers
1k views

x raised to itself infinite number of times [duplicate]

$$\Large x^{x^{x^{x^{x^{.^{\,.^{\,.}}}}}}} = 2$$ What is $x$?
1
vote
4answers
299 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 ...
1
vote
0answers
21 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
votes
1answer
175 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
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 ...
24
votes
2answers
835 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 ...
2
votes
0answers
112 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
votes
1answer
202 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
126 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 ...