# Tag Info

### Is there a generalization of factoring that can be extended to the Real numbers?

This is a question that is usually answered via Ring Theory. You can think of a ring as a set where addition, substraction, and multiplication is well defined. Here, we aren't thinking of ...
• 396

### Is there a generalization of factoring that can be extended to the Real numbers?

Obviously you can factor a real number into any set of numbers whose product is that real number. As for PRIME factorizations I know of one generalization of that probably DOESN'T work well for real ...
• 253

### Is there a generalization of factoring that can be extended to the Real numbers?

Let's take a different perspective than the other answers have taken so far. You can think of prime factorizations as telling you exactly how the natural numbers $\mathbb{N}$ (here I am excluding zero)...
• 431k
Accepted

### Comparing the dissimilarity of the order of integer sequences

There are various (Dis)Similarity measures. (A) You have given Dot Product , which is very common. $S=\Sigma a_ib_i$ (B) Commenter Malady has indicated Vector Norm , though there are variations there. ...
• 12.1k
Accepted

### Counting odd integers in a consecutive sequence divisible by a given prime

If I understand the problem correctly, the answer seems to be no, because $1$ is relatively far away from an odd multiple of a prime. For example: (With $p=7$, $a=6$, $n=19$): $O_7(1,1+19)=1$ ($7$ is ...
• 40.9k
1 vote
Accepted

### Two's complement multiplication decomposition

Let the binary form of $x \cdot y$ be represented using $\vec{a}$ whose dimension is $2w$. Therefore, by Equation 2.3  \begin{aligned} x \cdot y = B2T_w(\vec{a}) &\overset{\text{def}}{=} -a_{2w-...
• 591
1 vote

### Is there a generalization of factoring that can be extended to the Real numbers?

Any rational $x>0$ can be uniquely written as $x=\prod{p_i}^{m_i}$ with $p_i$ the $i^\text{th}$ prime and $m_i\in\mathbb Z$ its associated multiplicity. This extends the notion of factorization to ...
• 1,768
1 vote
Accepted

### Demonstration of the inverse additive of an element of Z is unique

If you are working in a structure that assumes commutativity as an axiom, then your proof is fine. You just found another way to prove the same statement. Otherwise, there are structures that do not ...
1 vote

### Axiomatic reason why $a=4 \implies a>1$ for $a \in \mathbb{N}$

The $>$ is not part of the language used in the Peano Axioms, so if we want to prove this statement, we'll need to add a definitional axiom. For example, we can use: \$\forall x \forall x (x > y \...
• 101k

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