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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Nested powers of $\sqrt 2$ has a solution different from its limit. What does this mean?

The infinitely nested power expression below has a limit of $2$: $$x=\sqrt2^{\sqrt2^{\sqrt2^{...}}}$$ In finding this limit, we may use: $$x=(\sqrt2)^x$$ But this expression has two solutions, $2$ ...
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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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Inverse and named fixed values, with ↑↑?

The inverse of $+$ is $-$, of $\times$ is $/$ and of $\text{^}$ is Log. Continuing upwards hyperoperationally, what is the inverse of $↑↑$? Whats more somtimes values that are fixed are given names, ...
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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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x raised to itself infinite number of times [duplicate]

$$\Large x^{x^{x^{x^{x^{.^{\,.^{\,.}}}}}}} = 2$$ What is $x$?
What is the area under the curve where the infinite power tower converges? $$\lim_{y \to \infty} = {}^y x.$$ The formula for this curve is given by various sources as: \frac{\mathrm{W}(-\ln x)}{-\...
$2^{3^{4^{…^{n}}}} \equiv 1$ (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 ...