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5
votes
0answers
506 views

Is integer factorization harder than RSA ($n=pq$) factorization? [closed]

This is a repost. I could not get a precise answer on math.SE and cstheory.SE Let FACT denote the integer factoring problem: given $n \in \mathbb{N},$ find primes $p_i \in \mathbb{N},$ and integers ...
2
votes
1answer
612 views

What are the chances of finding a small factor?

EDIT 2011.05.09 Thanks to Junkie and Tapio Rajala for checking on me. While most of the candidates referred to below have small factors, the "large" small factors I list below are incorrect. Also, ...
1
vote
2answers
422 views

when does a regular graph have a 1-factorization?

Is there a sufficient condition for a regular graph to have a 1-factorization (i.e. being able to pack all of its edges into disjoint perfect matchings, and excluding one vertex if the number of ...
0
votes
1answer
130 views

“locally” factoring subgroups of Lie groups

I'm not really a math person, and apologize if the question here is too simple. I've ended up with the following type of question for a few Lie groups, but state it for SO(n). I start with a subgroup ...
7
votes
3answers
705 views

Density of Irreducible Polynomials in $\mathbb{Z}[x]$

Recently I was thinking about some questions concerning $\mathbb{Z}[x]$ and realized that they might be a bit easier if I knew the relative densities of reducible polynomials. Let $P_d$ denote the ...
2
votes
2answers
472 views

Second stage of elliptic curve factorization via random walk/Pollard's rho in constant (or low) memory?

The second stage of elliptic curve factorization has the drawback of large memory usage. Let $n=pq$, $E(\mathbb{Z}/n\mathbb{Z})$ is elliptic curve and $P$ point on $E(\mathbb{Z}/n\mathbb{Z})$. On ...
1
vote
1answer
297 views

Cartesian factorization of a finite set of n-tuples

I'm interested in factoring a finite set of $n$-tuples as the Cartesian product of two "factor sets", of which the first factor is itself the Cartesian product of some of the set's projection images, ...
2
votes
1answer
641 views

A subring question (revised)

Hello, Let $K/{\mathbb Q}$ be a finite extension which is not necessarily Galois, and ${\mathcal O}$ be the ring of integers of $K$. Let $p$ be a prime in ${\mathbb Q}$ and let $p {\mathcal ...
7
votes
4answers
1k views

Consecutive numbers with n prime factors

Let $P(m,n)$ mean that there is a number, $M$, such that starting with $M$ there are $m$ consecutive numbers each having exactly $n$ distinct prime factors. Is it obvious that $P(m,n)$ is true for ...
0
votes
2answers
742 views

Prime factorization theory

Firstly, let me divulge. I've been doing a lot of research on the summation of two coprime numbers and unfortunately have failed to come up with the properties I'm seeking; it is my hope that someone ...
7
votes
1answer
188 views

Adding a multiple of the Identity to a LU factorized matrix

Suppose a square, dense, symmetric matrix $A$ has been factorized into $L$ and $U$ components by performing a LU decomposition. Now let $B = A+\lambda I$. Is there any way to efficiently compute the ...
6
votes
2answers
4k views

Fast trace of inverse of a square matrix

Which would be the most efficient way (in computational time) to compute tr(inv(H)), where H is a (dense) square matrix? In my particular problem I also have a LU decomposition of H already ...
0
votes
1answer
158 views

QR factorization: How to get decreasing r_ii

Hi, I'm attempting to implement a QR factorization with column pivoting so that the returned R matrix has decreasing diagonal elements (that is, $r_{i,i} \leq r_{i-1,i-1}$ for all $i\geq 2$). ...
2
votes
0answers
148 views

Pulling out factors in a Noetherian Domain

Let $R$ be a Noetherian domain (not-necessarily commutative), and let $S$ be a Noetherian subring of $R$. An element $r\in R$ is left $S$-irreducible if, for any $s\in S$ and $r' \in R$ with $sr'=r$, ...
3
votes
1answer
391 views

Factorization in the Wiener algebra on the unit disc.

Consider the Banach algebra $W^+=\ell^1(\mathbb{Z}^+)$, viewed upon as the analytic functions $f$ on the unit disc $\mathbb{D}$ such that $$\|f\|=\sum_{k\ge0}|a_k|<\infty$$ where $$f(z)=\sum ...
14
votes
2answers
1k views

How divisible is the average integer?

I don't know any number theory, so excuse me if the following notions have names that I'm not using. For a positive natural number $n\in{\mathbb N}_{\geq 1}$, define $Log(n)\in{\mathbb N}$ to be ...
4
votes
2answers
984 views

Factoring and solving trinomials

Has the problem of factoring (over the rationals) the general trinomial $ax^n+bx^k+c$ with $a,b,c\in\mathbb{Z}$, $n,k\in\mathbb{N}, n>k>1$ been solved? By solved I mean a classification theorem ...
12
votes
6answers
1k views

Seeking Noetherian normal domain with vanishing Picard group but not a UFD

Once again, the question says it all. My motivation is the article on factorization I am writing. I want to explain (as well as to understand!) why for normal Noetherian domains of dimension greater ...