Tagged Questions

If iterated exponentiation (usually also called "Powertower") is understood as mathematical operator or as function depending on the iteration height, then this operator/operation/function is often called "tetration" and is assumed as next step in the operator hierarchy addition,multiplication, ...

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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 ...
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Is the positive root of the equation $x^{x^x}=2$, $x=1.47668433…$ a transcendental number?

I can prove using the Gelfond–Schneider theorem that the positive root of the equation $x^{x^x}=2$, $x=1.47668433...$ is an irrational number. Is it possible to prove it is transcendental?
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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 ...
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How can I prove $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}…}}}}=2$ [duplicate]

How can I prove $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}...}}}}=2$$ I don't know which method can be used for this?
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What is $i$ exponentiated to itself $i$ times?

I was just wondering about this. I searched about it on the net and found that it is called tetration and after this comes pentation and then hexation and so on so forth. I don't really understand ...
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Proof of strictly increasing nature of $y(x)=x^{x^{x^{\ldots}}}$ on $[1,e^{\frac{1}{e}})$?

The title is fairly self explanatory: I have been trying to rigorously prove that $y(x)=x^{x^{x^{\ldots}}}$ is a strictly increasing function over the interval $[1,e^{\frac{1}{e}})$ for a while now, ...
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How to evaluate fractional tetrations?

Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the ...
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Prime factor of $2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4$

I checked the prime factors of $$2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4 = 2^{2^{2^2}} + 3^{3^{3^3}}$$ with trial division and found non below $8*10^9$ Nevertheless, the given number has still ...
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A puzzle with powers and tetration mod n

A friend recently asked me if I could solve these three problems: (a) Prove that the sequence $1^1, 2^2, 3^3, \dots \pmod{3}$ in other words $\{n^n \pmod{3} \}$ is periodic, and find the length of ...
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Complex towers: $i^{i^{i^{…}}}$

If $w = z^{z^{z^{...}}}$ converges, we can determine its value by solving $w = z^{w}$, which leads to $w = -W(-\log z))/\log z$. To be specific here, let's use $u^v = \exp(v \log u)$ for complex $u$ ...
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Is it possible to prove the positive root of the equation ${^4}x=2$, $x=1.4466014324…$ is irrational?

(somewhat related to my earlier question) Let ${^n}a$ denote tetration $\underbrace{a^{a^{.^{.^{.^a}}}}}_{n \text{ times}}$ (or, defined recursively, ${^1}a=a$, ${^{n+1}}a=a^{({^n}a)}$). The ...
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Derivative of $x^{x^{\cdot^{\cdot}}}$?

The infinite tetration is defined as $$f(x)=x^{x^{\cdot^{\cdot}}}$$ This function is defined for $e^{-e} \leq x \leq e^{e-1}$. (Wikipedia image) Can one determine the derivative of this function?
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Solve $x^{x^x}=-1$

I do know that if you use tetration the equation would look like this. $$^3x=-1$$ You could then theoretically use the super-root function to solve the equation, but I do not know how to use the ...
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Prime factors of $\sum_{k=1}^{30}k^{k^k}$

I checked the prime factors of $$\sum_{k=1}^{30}k^{k^k}$$ and did not find any upto $10^8$ Are there any useful restrictions to accelerate the search ?
How to calculate $f(x)$ in $f(f(x)) = e^x$?
How would I calculate the power series of $f(x)$ if $f(f(x)) = e^x$? Is there a faster-converging method than power series for fractional iteration/functional square roots?