# 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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### $x^{x^{x^{x^{x^{...}}}}} = 2$. 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, ...
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### 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 ...
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### 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 ...
1 vote
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### 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 ...
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1 vote
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### 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.
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\prod$ ...
1 vote
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### 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 ...
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### $x^{x^2} + x^{x^8} =?$ 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 ...
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### 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 ...
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### 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 : $$f''(x)\ge 0$$ ...
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### $f(n)=$, 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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....