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.

Filter by
Sorted by
Tagged with
9
votes
3answers
378 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 ...
2
votes
2answers
238 views

Do we know the value of $3 \uparrow\uparrow\uparrow 3$

I was studying Graham's number and before we can even start calculating $g_1$ which is: $g_1 = 3\uparrow\uparrow\uparrow\uparrow 3$, I was wondering if we even have the actual value of: $3 \...
1
vote
1answer
66 views

Defining the p-Frobenius for $GF(p^{16})$ explicit

Basics First let me tell you some background. Consider $p\equiv 13 \bmod 32$, therefore, $p\equiv 5\bmod 8$. That means, 2 is a none-quadratic residue in $GF(p)$. Therefore, we are able to build up $\...
1
vote
2answers
65 views

How does this equation follow from this?

Let $p>2$ be an odd number and let n be a positive integer. Prove that $p$ divides $$1^{p^n}+2^{p^n}+...+(p-1)^{p^n}.$$ Here is the solution: Define $k = p^n$ and note that $k$ is odd. Then \...
0
votes
2answers
146 views

Solve X=sqrt(A)^sqrt(A)^sqrt(A)^…infinty? [duplicate]

If $X= \newcommand{\W}{\operatorname{W}}\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{.{^{.^{\dots}}}}}}}}}}} $ then what is the value of $X^2-e^{1/X}$ ?
35
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 ...
0
votes
2answers
102 views

Difference between calculator and google calc for power [duplicate]

I tried to compute the power of 2^2^2^2 on google calculator and my casio calculator but both are giving different results. same is true for 3^3^3. Please explain me the difference between two ...
6
votes
0answers
81 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 ...
0
votes
2answers
312 views

Infinite Power Tower

I've been having fun with the problem of finding the values of $n$ for which the infinite power tower $$\sqrt{2}^{\sqrt{2}^{...^{\sqrt{2}^n}}}$$ Has a finite value. My final answer was that it ...
0
votes
0answers
442 views

Compute modulo of prime power tower

I have a prime number $p$, and I need to compute $p$ ↑↑ $k$ mod $m$. here p ↑↑ k can be written as p ^ (p ^ (p ^ (p ... k times))) for example - $p = 5, k = 3, ...
-1
votes
2answers
324 views

How to efficiently find a prime number $x$ raised to the power $x$ $k$ times modulo $m$?

How to efficiently find a prime number $x$ raised to the power $x$ $k$ times modulo $m$? In other words, how to find $ \underbrace{x^{x^{...^{x}}}}_k \mod m$, where $x$ is prime?
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?
1
vote
0answers
170 views

How can I solve this infinite exponent tower?

Using calculus (or algebra), how would I solve an infinite exponent tower such as this? $$c_0x^{c_1x^{c_2x^{c_3x^{.^{.^.}}}}}=a$$ Where $c_0=1$ and $c_{n+1}=\frac{c_n}{2}$ for $n=0,1,\ldots$ and $a&...
1
vote
1answer
266 views

Find the last ten digits of this exponential tower.

I came across this challenge. Find the last ten (least significant) decimal digits of $x$ = 2^(3^(4^(5^(6^(7^(8^9)))))). First some notation. Let $x_n = n^{(n+1)^{(n+2)^{.^{.^{.^9}}}}}$ denote ...
1
vote
4answers
813 views

Find the last two digits of $9^{9^{9}}$ [duplicate]

I want to find the last two digits of $9^{9^9}$ or $9^{81}$. I tried using Euler's theorem but I can't make anything of it. Any hint or a guide? Thanks!
0
votes
2answers
152 views

Confused with exponent rules?

What power would I need to raise $4^{2^{2^l}}$ to get $4^{2^{2^n}}$ where $n>l$? Very simple question but it has stumped me. I guess $2^{2^{n-l}}$ but I do not think that is right I am not sure?
2
votes
0answers
84 views

Fatou Coordinate with two complex conjugate fixed points and extending Tetration to real values

Lets us define $f(z)=z^2+z+c$ with real valued c>0 and iterate the function f(z). Then the Abel function for $f(z)$ is $$\alpha(z)\;\; \text{where}\;\; \alpha(f(z))=\alpha(z)+1$$ $$ f^{[\circ z]} = \...
2
votes
2answers
196 views

How to find $3^{3^{3^{\dots}}}\pmod{100}$?

I can show that $3^{3^{3^n}}\equiv7\pmod{10}$ since $3^1\equiv3\pmod{10}$ $3^2\equiv9\pmod{10}$ $3^3\equiv7\pmod{10}$ $3^4\equiv1\pmod{10}$ Thus, it reduces to $3^{(3^{3^n}\mod4)}$. I can then ...
3
votes
2answers
199 views

Power tower question

$$x^{x^{x^{.^{.^{.}}}}} = 8$$ Then how to solve for x? I first tried like this $x^8=8$ but I don't get any way to solve.
0
votes
6answers
235 views

Why is $2^{2^{2^n}}$ not equal to $16^n$?

Why is $a^{b^{c^d}}$ not equal to ${(a^{b^c})}^d$ (for positive n)? For example, WolframAlpha seems to say that $2^{2^{2^n}}$ is not equal to $16^n$.
5
votes
0answers
153 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 ...
6
votes
4answers
184 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, ...
5
votes
2answers
170 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
2answers
620 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 ...
2
votes
1answer
61 views

Evaluate $\frac{x^2}{y}$ where $x=a^{a^{a}}$ and $y=a^{a^{2a}}$

Evaluate $\frac{x^2}{y}$ where $x=a^{a^{a}}$ and $y=a^{a^{2a}}$ 1.$1$ 2.$x^{a^{a}}$ 3.$x^{1-a^a}$ 4.$x^{2-a^a}$ My solution: $x^2=a^{a^{a}}*a^{a^{a}}=a^{2a^{a}}$ $\frac{x^2}{y}=\...
3
votes
2answers
571 views

If $x^{x^{x^{16}}}=16$ calculate the value of $x^{x^{x^{12}}}$

I have done the following but I'not satisfied. if $x^a=a$ then by substitution follows that $x^a=x^{x^a}=x^{x^{x^a}}$ etc. So $x^{x^{x^{16}}}=16$ is equivalent to $x^{16}=16$, and $x=2^{\frac{1}{4}}$...
6
votes
3answers
438 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?
3
votes
2answers
316 views

Infinite power tower paradox with e and pi.

I was experimenting with Euler's Identity. If $e^{((\pi*i)/2)}$ is $i$, couldn't you recursively plug in $i$ into the expression. For example: $$e^{((\pi/2)*e^{((\pi/2)*e^{((\pi/2)*e^{((\pi/2)*e^{((...
2
votes
1answer
477 views

Fifth last digit of a huge number

How can I find the fifth last digit of $5^{5^{5^{5^5}}}$? I tried to evaluate $5^{5^{5^{5^5}}}\pmod {100000}$. But the exponent is so huge that I'm unable to evaluate it. Also, $(5,100000)=5$ , so $5$ ...
4
votes
3answers
185 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 $m\uparrow\uparrow(n+1) > n\uparrow\uparrow n$ ...
0
votes
2answers
255 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?
1
vote
1answer
161 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^{2^{2^{\dots}}}=e^{-W(-...
3
votes
1answer
88 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
161 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 \...
10
votes
2answers
311 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.
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,...
1
vote
2answers
245 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 ...
1
vote
0answers
97 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 = \sqrt{2}^{\sqrt{2}^{\...
4
votes
1answer
69 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 ...
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 ...
0
votes
0answers
114 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 $a^{b^t}=t$" ...
11
votes
1answer
163 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 ...
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) ...
5
votes
1answer
314 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^...
0
votes
0answers
24 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. ...
-1
votes
1answer
134 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
475 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 ...
116
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 ...
2
votes
1answer
267 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
210 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 $A256179$...