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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.

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3↑↑↑3= ? but with 10 instead of 3 ( approximation, order of magnitude )

3↑↑↑3= (or near) in power tower of 10 or in ( Knuth ) arrow ↑ notation of 10 to get a sense of it's order of magnitude; I grasp numbers more easily with 10 3↑↑↑3 being the first really huge number in ...
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Algorithm for comparing the size of extremely large numbers

Is there a simple algorithm to decide which of the numbers $$a \uparrow ^b c \text{ and } d \uparrow ^e f$$ is the bigger one ? Using the hyperoperation, the numbers can be denoted with $$H_{b+2}(...
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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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Hyperidentities and Clones, Trivial observation, commutativity

In the book Hyperidentities and Clones they (Denecke and Wismath) write: $xy \approx yx$, in other words $$F(x,y)=F(y,x)$$ considered as a hyperidentity implies $$x\approx y.$$ I would like to ...
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What is the time complexity of finding the most significant digit of $3\uparrow\uparrow n$?

What is the time complexity of finding the most significant digit of $3\uparrow\uparrow n$? I know we can find the least significant digits in constant time using modular arithmetic, is the most ...
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Notation for n-ary exponentiation

We have $n$-ary sums ($\sum$) and products ($\prod$). Is there an $n$-ary exponentiation operator? $$\underset{i=1}{\overset{n}{\LARGE{\text{E}}}}\, x_i = x_1 \text{^} (x_2 \text{^} (\cdots \text{^} (...
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Is there a notation for exponentiation analog to capital-sigma notation Σ for addition and capital pi Π notation for multiplications?

There are the following common notations: Sums: $$\sum_{i=2}^4 i = 2 + 3 + 4 = 9$$ Products: $$\prod_{i=2}^4 i = 2\times 3\times 4 = 24$$ Is there a (theoretical) one for: Exponentiation (...
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For which n does the nth “hyperoperation number” n[n]n begin with n in base 2?

If $[n]$ denotes the $n$th binary hyperoperation in the sequence $(+,\times,\uparrow,\uparrow\uparrow,...)$, then the following equality is readily verified for $n=1,2,3,4:$ $$n\,[n]\,n\ =\ (n_2...)...
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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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How to solve $ax^x+bx+c=0$?

How can I solve $$ax^x+bx+c=0$$ or $$ax^{x^x}+bx^x+cx+d=0$$ where $x^x$ and $x^{x^x}$ - tetration? Is there analogue of discriminant for it?
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Finite algebraic structures where all hyperoperations (addition, multiplication, exponentiation, tetration, etc.) are well-defined

Let $\langle R, +, \times, \uparrow, \uparrow\uparrow, \uparrow\uparrow\uparrow, \ldots; 0, 1\rangle$ be an algebraic structure with two constants $0, 1$ and where an infinite sequence of binary ...
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Are 'numerals' closed under exponentiation?

I have read Edward Nelson's Warning signs of a possible collapse of contemporary mathematics a couple of times, it is a very interesting read, but I do not understand the conclusory paragraph. In ...
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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 do hyperoperations like tetration exist if operations are seperate relations and not repeatitions of each other.

I've run into a bit of a conflict in my fundamental understanding of concepts in math. I've always known the arithmetic operation to be extensions of each other. Multiplication is repeated addition, ...
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Necessary conditions for being expressible using hyper-operators

We can recursively define expressions involving only addition and multiplication as follows: Let $K$ be a set of constant functions, usually the integers or some field. Functions in $K$ are defined ...
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Is there any concept similar to unique factorization that applies to exponential operators?

We can talk about prime numbers over multiplication but is there any similar concept that applies to exponential operators or other hyperpowers like tetration? Can we use what we know about UFDs to ...
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Can exponentiation and power function be defined through Albert Bennett's operations?

In 1914 Albert Bennett suggested the following operation: $$a * b=a^0_2b=\exp(\ln a \ln b)$$ Now, given this function, addition and multiplication, and their properties, can one express ...
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Can we evaluate noninteger hyperoparations?

The hyperoperation $a[n]b$ is addition when $n=1$, multiplication when $n=2$, exponentiation when $n=3$, tetration when $n=4$, and so on. What happens when $n$ is noninteger? Can we evaluate, e.g. $...
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Is there an equation $f(k,m,n)=n$ where $n$ is the number of applications of the Ackermann function needed?

Using the definition of the Ackermann function on page 247 of this paper, (sidenote, great paper): $$ \alpha(k,m,n) = \begin{cases} m+n, & k=1 \\ m, & n=1 \\ \alpha(k-1,m,\alpha(k,m,n-1)),&...
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Is tangent space a separable space constructed from a differentiable manifold as the algebraic surface between a plane and a hyperplane?

Entire question should be: Can we consider tangent space as the separable space constructed from a differentiable manifold as the algebraic surface between a plane and a hyperplane? Is possible to ...
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Generalized algebraic operations

I've looked around the web alittle, and besides hyper-operations it seems there has been little to no attempts at generalizing binary operations. Is there an operation space, where the operators exist....
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How are hyperoperations of rational and irrational numbers calculated?

Let's say I have the following problem, where x represents a real number. x = 2 * 3.4 I'm sure there are many ways one could solve this problem, but the way I would solve this is by adding 3.4 to 3....