# Questions tagged [tetration]

Tetration is iterated exponentiation, just as exponentiation is iterated multiplication. It is the fourth hyperoperation and often produces power towers that evaluate to very large numbers.

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### Are there integer solutions to the equation ${^n}a+{^n}b={^n}c$?

A couple days ago, someone posted a question about using integer solution to the equation $a^a+b^b=c^c$ to disprove Fermat's last theorem. The question has since been deleted but I was curious as to ...
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### Under what “natural” combinatorial conditions would tetration or higher hyperoperations appear?

This is quite a soft question and is not the same as this question. I want to know what sort of "natural" problems one might examine in combinatorics whose solutions naturally require higher order ...
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### Exponent is to exponentiation as _______ is to tetration

Would it be tetrand, tetrant, or something else?
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### Bounds of fractional tetration

I know about Kneser, but if we take a simple recursion $$^{1/d}b=c, a(0)=b, a(n)=b^{\frac{1}{^{d-1}(a(n-1))}}$$ so $$\lim\limits_{n\to\infty}a(n)=c$$ and we can quickly find $c$ for positive ...
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### How is tetration read in spoken English?

How would one read a tetration operation like $^4 3$ in spoken English? Meaning, what's the equivalent to reading $3 \times 4$ as "three times four" or $3^4$ as "three to the power of four" for ...
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### How to solve $x^x=a$ and related equations? [duplicate]

How can I solve the equation for $x$ when $x^x=2$ or any other constant? And is solving $x^{x^x}=a$ or $x^{x^{x^x}}=a$ or equations such as these even possible? What are these equations even called? ...
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### Regularization of Exponentially exponential series?

Question What are the convergence properties of the last equation: $$K = e^{x} + x + \ln{x} + \ln\ln(x) + \dots$$ Can one artificially choose a value of $\ln (x)$ (since there is more than one ...
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### What does $^3 3$ mean?

I was taking a test and this was a question. It does not mean $3^3$, nor is it a typo. The superscript was before the $3$ and I have no idea what it means. I tried researching but couldn't find an ...
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### $f(exp(z)) = f(z) \cdot g(z)$ with closed form solutions?

Are there closed form solutions for functions $f(z),g(z)$ Such that $A)$ $$\frac{f(exp(z))}{f(z)} = g(z)$$ $B)$ $g(z),f(z)$ are both analytic and nonconstant. $C)$ $g(z)$ is analytic at $z$ ...
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