# 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.

126 questions
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### How many values of $2^{2^{2^{.^{.^{.^{2}}}}}}$ depending on parenthesis?

Suppose we have a power tower consisting of $2$ occurring $n$ times: $$\huge2^{2^{2^{.^{.^{.^{2}}}}}}$$ How many values can we generate by placing any number of parenthesis? It is fairly simple for ...
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### On the power tower $\exp(x-\exp(x-\cdots))$

The intention is to find the maximum of the power tower $\exp(x-\exp(x-\cdots))$. From here, we see that it is around $0.965$ or possibly even higher. The approximate value of its integral is also ...
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### Partial Values for Knuth's Up-Arrows

$3 \uparrow 4$ is $3^4$, and $3\uparrow \uparrow 3$ is $3^{3^{3^3}}$, etc. For those of you unfamiliar, here is a wiki page on the notation. Clearly, up-arrow expressions, as they are usually ...
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### $2^{3^{4^{…^{n}}}} \equiv 1$ (mod $n+1$)

I remember when I started learning modular arithmetics I found a tetration equation stated as follows $2^{3^{4^{...^{n}}}} \equiv 1$ (mod $n+1$) I am wondering how could this be proved, I tried this ...
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### 3↑↑↑3= ? but with 10 instead of 3 ( approximation, order of magnitude )

3↑↑↑3= (or near) in power tower of 10 or in ( Knuth ) arrow ↑ notation of 10 to get a sense of it's order of magnitude; I grasp numbers more easily with 10 3↑↑↑3 being the first really huge number in ...
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### 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 $$g(x)=x!^{x!^{x!^{x!^{x!^{x!^{x!^{.{^{.^{.}}}}}}}}}}\implies g(x)=x!^{g(x)}$$ Taking natrual log on both ...
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### The Tower of 5 5s

What are the last three digits of $5^{5^{5^{5^5}}}$? I tried using modular arithmetic, but it had fallen short. A detailed solution is greatly appreciated. Thank you very much in advance!
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### $f(x) = \sqrt x^{{\sqrt[3]{x}}^{\sqrt[4]{x},\cdots}}$ asymptotic?

Consider for positive real $x$ : $$f(x) = \sqrt x^{{\sqrt[3]{x}}^{\sqrt[4]{x},\cdots}}$$ How does this function behave ? How fast does it grow ? Faster than any fixed iteration of exp sure, but ...
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### Magnitude of $f_3(n)$ compared to power towers of tens

In the fast growing hierarchy , the sequence $f_2(n)$ is defined as $$f_2(n)=n\cdot 2^n$$ The number $f_3(n)$ is defined by $$f_3(n)=f_2^{\ n}(n)$$ For example, to calculate $f_3(5)$, we have to ...
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### Passing that ball on, where does it end up in? [duplicate]

$10$ people are seating on chairs around a circular table. These chairs are marked in a clockwise manner. There is a ball on the man’s hand who is seated on $0$ marked chair, and the ball will be ...
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### Taylor series of a power tower

I recently proved that the Taylor Series of $\exp(\exp(x))$ is given by $$\exp(\exp(x))=\sum_{n=0}^\infty \frac{eB_n x^n}{n!}$$ where $B_n$ are the Bell Numbers. However, I can't figure out a Taylor ...