# Questions tagged [hyperoperation]

Hyperoperation is a field of mathematics which studies indexed families of binary operations, Hyperoperations families, that generalize and extend the standard sequence of the basic arithmetic operations of addition, multiplication and exponentiation.

93 questions
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### Why are addition and multiplication commutative, but not exponentiation?

We know that the addition and multiplication operators are both commutative, and the exponentiation operator is not. My question is why. As background there are plenty of mathematical schemes that ...
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### Example of Tetration in Natural Phenomena

Tetration is a natural extension of the concept of addition, multiplication, and exponentiation. It is quite obvious that there are things in the physics world which can be modeled by these 3 lowest ...
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### Has this phenomenon been discovered and named?

If $$x-\frac{x}{2}=\frac{x}{2},$$ and $$\frac{x}{\sqrt{x}}=\sqrt{x},$$ and $$x-\uparrow(x-\uparrow^22)=x-\uparrow^22$$ when $(x\uparrow^n-[A])\uparrow^nA=x$, where $A$ is some constant, and one uses ...
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### Are there nontrivial equations for hyperoperations above exponentiation?

A similar question was asked in comments elsewhere. A paper by Roberto Di Cosmo and Thomas Dufour ("The Equational Theory of 〈ℕ, 0, 1, + , ×, ↑〉 Is Decidable, but Not Finitely Axiomatisable") asserts ...
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### How to define $A\uparrow B$ with a universal property as well as $A\oplus B$, $A\times B$, $A^B$ in category theory?

In category theory there are definitions for $A\oplus B$, $A\times B$ and $A^B$ via universal properties. I wonder if it is possible to isolate a particular universal property to represent the ...
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### Transfinite Knuth-arrow hierarchy vs. fast-growing hierarchy

Suppose Knuth arrow notation (and hence the hyperoperation sequence) is extended to transfinite ordinal indices as follows: Let μ be a large countable ordinal such that a fundamental sequence is ...
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### Are hyperoperators primitive recursive?

I apologize if this question is too basic. I have read that the Ackerman function is the first example of a computable but NOT primitive recursive function. Hyperoperators seem to be closely related ...
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### Ordinal tetration: The issue of ${}^{\epsilon_0}\omega$

So in the past few months when trying to learn about the properties of the fixed points in ordinals as I move from $0$ to $\epsilon_{\epsilon_0}$ I noticed when moving from $\epsilon_n$ to the next ...
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### Are points on the complex plane sufficient to solve every solvable equation composed of the hyperoperators, their inverses, and complex numbers?

Some background: I'm programming a maths environment. I'm computer science, so please excuse any probable ignorance and lack of precision in my question. It seems $i$ and complex numbers were "...
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### With $f(n) = n!$, what is the least $k$ such that $f^k(\text{googolplex}) > \text{Graham's number}$?

$\text{googolplex} = 10^{(10^{100})}$ Is $\text{googolplex}!$ greater than $\text{Graham's number}$? How would this be proven? If $\text{googolplex}! \le \text{Graham's number}$, (which I expect) ...
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### What is the geometric, physical or other meaning of the tetration?

What is the geometric, physical or other meaning of the tetration or more high hyperoperations? Is it exists in general or it has only math concept?
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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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### 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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### What can we say about the prime factors of $​^{10}10+23$?

In a video on ultrafinitism I saw a claim that the number $​^{10}10+23$ does not have prime factorization. While I don't accept the premise of ultrafinitism, I got curious, what can we say about the ...
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### Integrate $x$ to the power $x$… to the power $x$… infinitely

This came across my mind, integrating $x$ to the power $x$ infinitely, I couldn't find anything on it. $$\Large \int x^{x^{x^{x\,\cdots}}} \, dx$$ How would you go about this?
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### Question about $TREE(3)$ and Graham's Number

Lets say: $G=\text{Graham's Number}$. And:  \begin{align*} \alpha_1&=G\uparrow^G G, \\ \alpha_2&=\alpha_1\uparrow^{\alpha_1} \alpha_1 \\ &\vdots\\ \beta_1 &= \alpha_G\uparrow^{\...
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### Does domination of exponential-factorial by tetration generalize to higher-order hyperoperations?

Let $\star$ be any operation in the sequence of hyperoperations $(\text{Succ},+,\times,\uparrow,\uparrow\uparrow,\ldots)$, and consider the $\star$-factorial function defined as follows on the ...
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### Is there a mathematical term, practical application, or area of math that covers a function raised to itself?

Some abstract examples would be: $f(x)^{f(x)}$ or $f(x)^{f(x)^{f(x)...}}$ Actual equations I've attempted to look at can be viewed here on desmos.com There seems to be a pattern of common convergence, ...
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### How much bigger is 3↑↑↑↑3 compared to 3↑↑↑3?

3↑↑↑3 is already mind-bogglingly large, but how much larger is 3↑↑↑↑3? Is it so large that it is simply around 3↑↑↑↑3 times larger than 3↑↑↑3? Or is there another way to express its magnitude in terms ...
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### Can $10\uparrow^n m<2\uparrow^n (m+2)$ be formally proven?

See here : http://googology.wikia.com/wiki/Arrow_notation for the definition of the up-arrow function. Can $10\uparrow^n m<2\uparrow^n (m+2)$ be formally proven for all $m\ge 1$ and $n\ge 3$ ?...
### Hyperoperation: Why does $H_n(0,b) = 0$ for $n\ge4$, $b$ odd ($\ge -1$)
On the Wikipedia for hyperoperations is says that $H_n(0,b) = 0$ if $n\ge4$, $b$ odd ($\ge -1$). From what I gether, the basic jist of what this is saying is that $0^{0^0}=0$ and from that we can ...