Skip to main content

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
13 votes
1 answer
640 views

$x^{x^{x^{x^{x^{...}}}}} = 2$. Why is $-\sqrt{2}$ not a solution?

I just watched a video by blackpenredpen where he solved this equation. Here are the steps he took (The process transitions from one line to another. He didn't explicitly use an implies or iff sign, ...
ten_to_tenth's user avatar
  • 1,426
4 votes
1 answer
307 views

Easy proof that $n=5$ is the only solution of $n^n \equiv n^{n^n} \pmod {10^{n-1}}$ if $n \in \mathbb{N}-\{0,1\}$ is not a multiple of $10$

Let $n > 1$ be an integer not a multiple of $10$. Is there a short proof that $n = 5$ is the only solution to $n^n \equiv n^{n^n} \pmod {10^{n-1}}$, given the fact that $5^{5^5} \equiv 3125 \pmod{...
Marco Ripà's user avatar
  • 1,164
0 votes
2 answers
98 views

Show that the number $2^{2^{2n+1}}+2^{2^{2n}}+1$ is not prime.

question Show that the number $2^{2^{2n+1}}+2^{2^{2n}}+1$ is not prime, for any nonzero natural number $n$. my idea $ 2^{2^{2n+1}}+2^{2^{2n}}+1 = 2^{2^{2n}}*5+1 $ i did this by giving common factor. ...
IONELA BUCIU's user avatar
1 vote
0 answers
86 views

Intriguing properties of power tower function

I was talking with an acquaintance and he proposed an interesting function: Consider a number $x$. Take the fraction $\frac{x}{x - 1}$, and raise it to the power $\frac{3x}{3x - 1}$. Iterate this ...
trebledawson's user avatar
2 votes
0 answers
194 views

Find the value of $\frac{1}{2}^{\frac{1}{3}^{\frac{1}{4}^{\cdots \infty}}}$? [duplicate]

Greetings with utmost respect, everyone! Today, I found a fascinating math question online. I seem to be stuck while solving it, however. I did not find any relevant solution to the problem yet. ...
Rohan Bari's user avatar
5 votes
0 answers
143 views

Solve for $m(t)$ in the integral transform $\int_0^1(1-t^n) m(t) dt=\frac{(n+1)^n}{n^{n-1}} $ for $n>0$.

Background (You can skip this part, but maybe you find it interesting.) Is $ \displaystyle f_1(x,v) = \sum_{n=0}^{\infty} \frac{x^n}{(n!)^v } > 0 $ for all real $x$ and $0<v<1$ ? Lets start ...
mick's user avatar
  • 16.4k
2 votes
1 answer
123 views

Numbers such that $ \sqrt{ {a_{1}}^{a_{2}^{...^{a_{n}}}}} = a_{1} a_{2} \dots a_{n}$

I wonder whether there are any resources on the equation: $$\require{\MnSymbol} \require{\mathdots} \sqrt{ {a_{1}}^{a_{2}^{...^{a_{n}}}}} = a_{1} a_{2} \dots a_{n}. \tag{1}$$ Here, $a_{1}, a_{2}, \...
Max Muller's user avatar
  • 7,208
4 votes
0 answers
306 views

Convergence of Variable Base Power Tower $\frac{1}{2}^{\frac{1}{3}^{\frac{1}{4}^{\ldots}}}$

I’m curious if there is a heuristic method that can be used to solve what appears to be an elementary power tower where every increasing power is decreasing by a half integer. Numerical methods ...
Kyler Rusin's user avatar
0 votes
0 answers
81 views

Infinite power tower for $x > e^{1/e}$

For $x>0$, let $\tau_0 = 1$ and $\tau_{n+1} = x^{\tau_n}$. The infinite power tower of $x$ is then $\tau = \tau(x) = \lim_{n\to \infty} \tau_n$. It is well known that $\tau$ exists and is finite ...
rubiko's user avatar
  • 1
1 vote
1 answer
167 views

Solve the tower equation: $\displaystyle x^{x^{x^{x+1}+x+1}}=2$

We are struggling to solve a crazy looking equation below: $$\displaystyle x^{x^{x^{x+1}+x+1}}=2$$ Approximate numerical values ​​are unfortunately not a solution. Wolfram Alpha offers only the ...
mathtime's user avatar
1 vote
1 answer
67 views

Can I find $x^{x^{-1}}$ given $x^{-17^{ 17^{-x}}}=17$?

Has This problem a solution? I tried for many ways, even Mathematica. I think it has a typo: $$x^{-17^{ 17^{-x}}}=17$$ Find: $$x^{x^{-1}}$$ How can I explain it has no solution. Kind regards.
Fernando Antonio Ponce's user avatar
10 votes
1 answer
380 views

How to solve the ODE $(y(x))^{((y'(x))^{((y''(x))^{((y'''(x))^{\cdot^{\cdot^{\cdot}}})})})}=f(x)$?

Introduction Why This Question? I have often asked myself what a solution to the Ordinary Differential Equation (ODE) would be and when I recently saw this ODE again, tried it again and failed again, ...
Kevin Dietrich's user avatar
3 votes
3 answers
99 views

If $\lim_{n\to \infty} a_n = 4$, can we say $\lim_{n\to \infty} a_{n-2} = 4$?

I was doing this exponential tower equation: $$2^{x^{2^{x^{2^{...}}}}} = 4$$ $$\text {(each new exponent is the power of the last exponent)}$$ The popular method is to break the tower at the first ...
ten_to_tenth's user avatar
  • 1,426
13 votes
1 answer
417 views

Closed form of $\begin{align}\int_{0}^{\infty}x^{1-x^{2-x^{3-...}}}dx\end{align}$.

I want to find the closed form of $\begin{align}\int_{0}^{\infty}x^{1-x^{2-x^{3-...}}}dx\end{align}$. Quick disclaimer: I have no reason to believe one actually exists Using Desmos, the closest I have ...
Dylan Levine's user avatar
  • 1,724
0 votes
2 answers
232 views

How to solve $x^{y^z}=z$

Initially I isolated the y in $x^y=y$, but I just wanted to expand the infinite power tower to two letters in the tower, but I can't solve for z in the equation $x^{y^z}=z$. I tried to use Lambert W ...
Tio Zuca's user avatar
  • 380
6 votes
1 answer
116 views

Operation that Turns Powers into Products Like How Logarithms Turn Products into Sums

$$ \newcommand{\pow}{\mathop{\vcenter{\huge{\text{E}}}}\limits} $$ The $\sum$ operator can be defined recursively as $$ \sum_{i = a}^b f(i) = f(a) + \sum_{i = a + 1}^b f(i). $$ Likewise, the $\prod$ ...
William Ryman's user avatar
1 vote
1 answer
187 views

Power tower modulus

I'm doing a programming challenge and I can't wrap my head around of finding an Euler totient when the modulus and the base aren't co-primes. So I have: $$4 ^ {4 ^ 4}(mod\;10)$$ I understand that 4 ...
CaseMon's user avatar
  • 31
2 votes
2 answers
166 views

i raised to itself

Last night I learned the amazing fact that $i^i=e^{-\pi/2}$ So I started computing other powers and would like to get confirmation that: $i^{i^i}=e^{-i\pi/2}$ $i^{i^{i^{i}}}=e^{\pi/2}$ $i^{i^{i^{i^{i}}...
Sydney Carton's user avatar
0 votes
1 answer
83 views

Power rules: $x ^ {(y^z)}$

I know about $(x^y)^z$ = $x^{yz}$ but what about $x^{(y^z)}$? Are there any rules for this? Let's consider the power of power rule: $(x^y)^z$ = $x^{yz}$: $(2^3)^4$ = $8^4$ = 4096 $2^{3\times 4}$ = $2^...
Rugved Modak's user avatar
2 votes
3 answers
139 views

$x^{x^2} + x^{x^8} =?$ Given $x^{x^4} = 4$

If x is any complex number such that $x^{x^4} = 4$ , then find all the possible values of : $x^{x^2} + x^{x^8}$ First, I used laws of exponents to give $18$ as answer. However , I realised that I've ...
Get_ Maths's user avatar
1 vote
2 answers
215 views

Find $f$ such that $f(f(x))=\log (x)$

Which is the solution for the following expression $$f(f(x))=\log (x)?$$ In other words, which function composed with itself matches with logarithm? With $x\in (0,\infty )$. The breakdown of ...
Francesco Bianco's user avatar
2 votes
0 answers
40 views

Stack the digits of number

$0^0$ is defined as $1$ in this question. 1.I have a natural number $N$, $N=\overline{a_1a_2a_3\cdots a_n}$, where $\overline{a_1a_2a_3\cdots a_n}$ denotes the number in decimal representation. 2.If $...
SegmentTree's user avatar
0 votes
1 answer
41 views

Digits tower power iterate

Stack the digits of a natural number into a power tower, iterate until only one digit remains. Does this iteration always terminate for any positive integer? Additionally specify $0^n = 0$, even when $...
Aster's user avatar
  • 1,230
-2 votes
1 answer
145 views

Correct my sketch of proof about the convexity of the "natural" power tower on $[1,\infty)$

Hi I want to show the following fact : Problem : Let $x\geq 1$ and $n\geq 1$ a natural number and define: $$f(x)={}^{2n}x=\underbrace{x^{x^{⋰^{x}}}}_{2n\text { times}}$$ Then we have : $$f''(x)\ge 0$$ ...
Ranger-of-trente-deux-glands's user avatar
3 votes
1 answer
96 views

$f(n)=$, for even integer’s $n$, $\lim_{n\to\infty} \int_{0}^{\infty} \frac{e^x}{^nx} dx$

$f(n)=$, for even integer’s $n$, $$\lim_{n\to\infty} \int_{0}^{\infty} \frac{e^x}{^nx} dx$$ $^n x$ is the tetrative function or ‘power tower’, which when $n=\infty$ can also be written in terms of y, ...
Maxwell Mogadam's user avatar
47 votes
4 answers
2k views

Is each of $\int_0^\infty\frac{dx}{x^x},\int_0^\infty\frac{dx}{x^{x^{x^x}}},\int_0^\infty\frac{dx}{x^{x^{x^{x^{x^x}}}}},\cdots$ less than $2$?

A few years ago I asked about the inequality Prove that $\int_0^\infty\frac1{x^x}\, dx<2$. As I came back to revisit it, I found that each of the following tetration integrals $$\int_0^\infty\frac{...
TheSimpliFire's user avatar
  • 27.1k
-1 votes
1 answer
128 views

Examples of closed forms of integrals with a power tower argument using W-Lambert function.

Here is a closed form of an integral that looks like: Integral form(s) of a general tetration/power tower integral solution: $$\sum\limits_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(an+b,cn+d)}{Γ(An+B)}$$ ...
Тyma Gaidash's user avatar
1 vote
2 answers
148 views

Evaluating negative infinite tetration: $\lim\limits_{n\to-\infty}\,^n x=\lim_{n\to\infty}\underbrace{\log_x\left(…\log_x(0)\right)}_n=\,^{-\infty} x$

One can learn that for a power tower of height $n$: $$y=n\big\{x^{x^{x^…}}=\,^nx\implies \log_x(y)=\boxed{\log_x(\,^nx)=\,^{n-1}x}$$ giving a recursive relation. One might see that $n<-2$ cases are ...
Тyma Gaidash's user avatar
4 votes
2 answers
201 views

Could someone explain why $\sum_{\substack{a_1,\ldots,a_n\in\mathbb{N}_0\\a_1+\cdots+a_n=n}}\frac{n!}{a_1!\cdots a_n!}=n^n$?

I want to use this equality but I have no idea why it holds. Sure I can probably prove it via induction but it looks rather fiddly. (Let $n$ be a positive integer.) $$\sum_{\substack{a_1,\ldots,a_n\in\...
UnsinkableSam's user avatar
2 votes
2 answers
124 views

About the inequality $f(-x)<\left(\Gamma(1-x)\right)^{\frac{1}{1-x}}$

Prove or disprove that $-1< x<0$ then we have : $$f(-x)<\left(\Gamma(1-x)\right)^{\frac{1}{1-x}}$$ Where : $$f(x)=2-x^{-\frac{212}{1000}x^{x^{\frac{1}{5}x^{2x^{\frac{1}{5}x}}}}}$$ My ...
Ranger-of-trente-deux-glands's user avatar
7 votes
1 answer
416 views

Can $\int\limits_0^\infty e^{ix^ x}dx$ be written without a limit?

We know about the Fresnel Integrals: $$C(x)=\int \cos x^2 \, dx,\quad S(x)=\int \sin x^2 \, dx$$ which can also be written as: $$\int e^{ix^2}dx=C(x)+i\,S(x)$$ To make a more interesting and tetration ...
Тyma Gaidash's user avatar
8 votes
1 answer
446 views

Integral form(s) of a general tetration/power tower integral solution: $\sum\limits_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(An+B,Cn+D)}{Γ(an+b,cn+d)}$

In many tetration/power tower integrals, one sees a general form of the following. Let this new function be notation used to show the connection between the general result and special cases using ...
Тyma Gaidash's user avatar
2 votes
0 answers
89 views

Is there any formula to sum the series $\sum_{k=1}^n{2^{k^k}}$

$$\sum_{k=1}^n{2^{k^k}}$$ I couldn't find anything on this with a simple Google search so is it not trivial? If yes then can someone link some resources/Wikipedia page where I could read more? (can't ...
Amadeus's user avatar
  • 1,107
7 votes
0 answers
375 views

Does $\mathrm{\int W(ln(x))dx}$ have a closed form?

This is follow up to this question which you will have to see for context: Is there a better solution for $$\mathrm{\int (a^t)^{(a^t)}dt= C+t+\frac1{ln(a)}\sum_{n=0}^\infty \frac{(-1)^n Q(n+1,-nt\,ln(...
Тyma Gaidash's user avatar
0 votes
2 answers
234 views

Solution to the Equation $x^x = 0$.

If I solve the Equation $x^x = 0$, then I get $x^x=0$: x ln⁡(x)= -∞ or ln⁡x $e^{\ln⁡(x)}$ =-∞ and if we take the Lambert W function on both sides we get ln⁡(x)=∞, And I put it on Wolfram Alpha then it ...
A High School Student's user avatar
4 votes
1 answer
512 views

Prove or disprove that the function is convex .

It seems we have : Define $\displaystyle f(x)=\sum_{k=1}^{2n}x^{k^2}$ where $n\geq 1$ a natural number and $-1\leq x\leq 1$ Claim : $f''(x)\geq 0$ My attempt : The case $n=1$ is trivial . So I have ...
Ranger-of-trente-deux-glands's user avatar
4 votes
2 answers
502 views

Correct my proof about : $x^{x^{x^{x^{x^{x}}}}}$ is convex for $0.25<x<0.27$

Let $0.25<x<0.27$ then define : $$f(x)=x^{x^{x^{x^{x^{x}}}}}$$ Claim: The second derivative of $f(x)$ is strictly positive . Proof : Let $0.25<x<0.27$ then define : $$g(x)=x^{x^{x^{x}}}$$ ...
Ranger-of-trente-deux-glands's user avatar
5 votes
1 answer
379 views

Is there a better solution for $\mathrm{\int (a^t)^{(a^t)}dt= C+t+\frac1{\,ln(a)}\sum_{n=0}^\infty \frac{(-1)^n Q(n+1,-nt\ln(a))}{n^{n+1}}\,dt}$?

I know there exist functions like this one for simplifying tetration based sums. There may be a way to simplify this type of sum at least using a lesser known and widely accepted functions. Here are ...
Тyma Gaidash's user avatar
4 votes
1 answer
244 views

How to evaluate the finite power tower $\tan(1°)^{\tan(2°)^{\tan(3°)^{\cdot^{\cdot^{\cdot^{\tan(44°)^{\tan(45°)}}}}}}}$

Consider the following finite power tower: $$\Large \tan(1°)^{\tan(2°)^{\tan(3°)^{\cdot^{\cdot^{\cdot^{\tan(44°)^{\tan(45°)}}}}}}}$$ I'm wondering if there is a way to solve this that doesn't rely on ...
Aussie Mathematician's user avatar
6 votes
1 answer
204 views

Is the infinite product $\prod_{i=0}^{\infty}(1+\frac{1}{2^{3^i}})$ transcendental?

Is the following number algebraic or transcendental? $$P:=\prod_{i=0}^{\infty}\left(1+\frac{1}{2^{3^i}}\right)$$ We could also define it as follows: let A be the set of natural numbers which contain ...
Riemann's user avatar
  • 727
1 vote
1 answer
237 views

On $\int \sqrt[x]x$dx. Solution found for $eW\left(\frac1e\right)\le x\le e$.

Notice: Note that there is hope for a more general answer because an answer for the area under the infinite tetration/power tower curve was almost certainly found. It uses the nice OEIS A008405 find. ...
Тyma Gaidash's user avatar
14 votes
1 answer
688 views

About the inequality : $x^{x^{x^{x^{x^x}}}}\geq x^{x^{x^{((e-2)(1+e))x\left(1+\sqrt{x}\left((\sqrt{x})^3-1\right)\right)}}}\geq x^{x^{\frac{16}{27}}}$

This inequality is due to user RiverLi : Let $0<x\leq 1$ then we have : $$x^{x^{x^{x^{x^x}}}}\geq x^{x^{\frac{16}{27}}} \geq 0.5x^2+0.5$$ I propose another one wich states : Let $0<x\leq 1$ ...
Ranger-of-trente-deux-glands's user avatar
3 votes
0 answers
130 views

Stronger statement : $^{6}x\geq e^{\left(\frac{\ln^{2}\left(x+1\right)}{\ln\left(2\right)}-\ln\left(2\right)\right)}\geq \left(x^{2}+1\right)0.5$

Let $x>0$ then prove or disprove that : $$x^{x^{x^{x^{x^{x}}}}}\geq e^{\left(\frac{\ln^{2}\left(x+1\right)}{\ln\left(2\right)}-\ln\left(2\right)\right)}\geq \left(x^{2}+1\right)0.5$$ My attempt : ...
Ranger-of-trente-deux-glands's user avatar
16 votes
2 answers
1k views

Integral over domain of infinite tetration of x over extended domain from 0 to $\sqrt[e]e$. Possible $\int_{e^{-e}}^{e^\frac1e} x^{x^{…}}dx$ solution.

I have been trying to find an interesting constant over the domain of the infinite tetration of x and have just almost figured out the area with a non integral infinite sum representation. Just one ...
Тyma Gaidash's user avatar
20 votes
3 answers
928 views

Prove or disprove that the function $f(x)=x^{x^{x^{x}}}$ is convex on $(0,1)$

Let $0<x<1$ and $f(x)=x^{x^{x^{x}}}$ then we have : Claim : $$f''(x)\geq 0$$ My attempt as a sketch of partial proof : We introduce the function ($0<a<1$): $$g(x)=x^{x^{a^{a}}}$$ Second ...
Ranger-of-trente-deux-glands's user avatar
8 votes
2 answers
714 views

Area under $x^{-x}$ over its real domain. What is another non-integral form of $\int_{\Bbb R^+}x^{-x}dx$?

A few years ago, I got interested in an apparently hard integration problem which had me fascinated. This was the integral of a Sophomore Dream like integral except with the bounds over the real ...
Тyma Gaidash's user avatar
9 votes
1 answer
398 views

Limits of negative-power tower

Consider the following: $$n \uparrow -\Bigl((n+1) \uparrow -\bigl((n+2) \uparrow \cdots \uparrow -m \bigr)\Bigr)= n^{{{-(n+1)}^{-(n+2)}}^{\cdots^{-m}}}$$ It doesn't converge for $m \to \infty$, but ...
Christian's user avatar
  • 2,125
2 votes
1 answer
65 views

If $\displaystyle x^{x^9}=\sqrt{3^{\sqrt{3}}}$ and $\displaystyle y=x^{\left(\frac{1}{y^{y^x}}\right)}$, determinate the value of $y^{3x}$.

If $\displaystyle x^{x^9}=\sqrt{3^{\sqrt{3}}}$ and $\displaystyle y=x^{\left(\frac{1}{y^{y^x}}\right)}$, determinate the value of $y^{3x}$. My try It is easy to see that if we raise the first equation ...
Trobeli's user avatar
  • 3,312
1 vote
1 answer
93 views

did I make a mistake? when trying to find the derivative of $x^{x^{x^{x^{x^{\dots}}}}}$

did I make a mistake? when trying to find the derivative of $f(x)=x^{x^{x^{x^{x^{\dots}}}}}$ the first thing I did was to change the $f(x)$ to this $$y=x^y$$ I took the derivative of both $y$ and $x$ ...
user avatar
0 votes
0 answers
51 views

Representing a Number as the Sum of Powers of Form $k^k$.

So I was wondering if it would be useful to instead of writing a number in base $2$ or $3$, we use functions in general as bases. So like writing it as the sum of squares or other increasing functions....
snowball's user avatar
  • 159

1
2 3 4 5