# 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.

125 questions
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### Pattern in power towers of 2 involving last digits

We have \begin{align} 2^{2^{2}} &\mod 10 = 6 \\ 2^{2^{2^2}} &\mod 100 = 36 \\ 2^{2^{2^{2^2}}} &\mod 1000 = 736 \\ 2^{2^{2^{2^{2^{2}}}}} &\mod 10000 = 8736 \\ 2^{2^{2^{2^{2^{2^2}}}}} &...
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### $f(x) = \sqrt x^{{\sqrt{x}}^{\sqrt{x},\cdots}}$ asymptotic?

Consider for positive real $x$ : $$f(x) = \sqrt x^{{\sqrt{x}}^{\sqrt{x},\cdots}}$$ How does this function behave ? How fast does it grow ? Faster than any fixed iteration of exp sure, but ...
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### The sequence satisfying $a_2^{a_3^{\dots^{a_n}}} = n$

$a_2 = 2$ $a_2^{a_3 } = 3$ So $a_3= \ln(3)/\ln(2)$. I wonder about all solutions $a_n$ such that $a_2^{a_3^{\dots^{a_n}}} = n$ For all $n$. How does $a_n$ behave? What are the best asymptotics? ...
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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 ...
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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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### 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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### 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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### 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 \...
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### Power Towers, and Notation for Iterated Exponentiation

So far, we use the symbol $$\sum$$ to denote sums, and $$\prod$$ to denote products. But is there any such notation for exponentiation? Has any research been done about exponentiation of this type, ...
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### 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 $X^2-e^{1/X}$ ?
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### 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 ...
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### 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 ...
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### A new interesting pattern to $i↑↑n$ that looks cool (and $z↑↑x$ for $z\in\mathbb C,x\in\mathbb R$)

Many of you may recall "An obvious pattern to $i↑↑n$ that is eluding us all?", an old question of mine, and just recently, I saw this new question that poses a simple extension to tetration at non-...
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### 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 ...
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### 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?
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### 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 ...
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### 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.
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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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### 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 $16^n$.
### 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 ...