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# 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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### 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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### Prove that $F_{2n}(x)=0$ has exactly one root in the interval $x\in(0,1),$ and this root $\to 0$ when $n \to \infty.$

Define $$f_1(x)=x\\f_2(x)=x^x\\\vdots\\f_{n+1}(x)=x^{f_n(x)}$$ Let $F_n(x)=f_n^{'}(x).$ Hence $$F_1(x)=1\\F_2(x)=x^x(1+\log(x))\\\vdots$$ Prove that $F_{2n}(x)=0$ has exactly one root in the ...
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### Solving $x^{1/x}=y^{1/y}$ iteratively by power towers, convergence

Suppose we have no idea that $x^{1/x}=\exp \left( \frac{1}{x} \ln x \right)$ or don't know anything about exponents and logarithms. But we can certainly compute roots, for example by this method. ...
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### Order of Recursion?

Define an extended algebraic function $f(a)$ as a function on $a$ that utilizes any combination of recursive extensions and inverses of sequentiation. Example: $a + 1$ , sequentiation. $a + a$, ...
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### Tetration Binomial Theorem

I was exploring tetration and came across the following identities: $${^0}(ab) = 1$$ $${^2}(ab) = ({^2}a)^b * ({^2}b)^a$$ $${^3}(ab) = (ab)^({^2}(ab)) = (ab)^{(({^2}a)^b * ({^2}b)^a)}$$ That third ...
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### Comparison of the Sizes Arbitrarily Generated Compound Exponential Numbers

I've been wondering about how it might be possible, given two compound exponential numbers generated by recurrence schemes equipped with addition multiplication and exponentiation (to some integer ...
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### Exponent is to exponentiation as _______ is to tetration

Would it be tetrand, tetrant, or something else?
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### another number group?

I noticed that for each basic increasing binary function (addition, multiplication, and exponentiation) its inverse (or just a inverse) of certain values adds more number types to the number line (or ...
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### Prove that ${}^ni$ is complex for all $n \ge 3$

We can define ${}^nx$ as $\underbrace{x^{x^{\cdot ^{\cdot ^{x}}}}}}_{n\text{ times}$ (Tetration). I conjecture that ${}^ni$ is complex for all $n \ge 3, n \in \mathbb{N}$. I've ...
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### Find all ordered triples of primes $(a, b, c)$ such that $a \mid {^bc} + 1, b \mid {^ca} + 1, c \mid {^ab} + 1$.

Find all ordered triples of primes $(a, b, c)$ such that $$\large a \mid {^bc} + 1, b \mid {^ca} + 1, c \mid {^ab} + 1$$ Notation: $$\large {^xy} \text{ or } x^{\underline y}$$ are nonations for ...
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### Proving the derivation of the infinite tetration via Summation/Product notation is equivalent to the derivation via Implicit Differentiation

First some background info. Feel free to skip to the bold text below if you already know this. I) The derivative of the infinite tetration of $x$ (I guess that's how you say it) or ${}^\infty x$ ...
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 ...
First I introduce a notation similar to $\sum_{i=1}^n a_i$ for exponentiation. I.e. for any (potentially infinite) sequence $a_i$ we define  ES_{i=1}^n a_i = \left\{\begin{matrix} a_1 &...