The factorization tag has no usage guidance.

**5**

votes

**1**answer

310 views

### Bounded domain matrix factorization

Assume we're trying to find $A\in [-1,1]^{n\times d}, B\in [-1,1]^{m\times d}$, from an observed matrix $C\in [-d,d]^{n\times m}$, where $C=AB^T$.
The goal is to return $\widehat A, \widehat B$ such ...

**11**

votes

**0**answers

325 views

### Do the coefficients of these irreducible polynomials always become periodic?

Fix $n\in\mathbb N$ and a starting polynomial (or seed) $p_n=a_0+a_1x+\dots+a_nx^n$ with $a_k\in\mathbb Z\ \forall k$ and $a_0a_n\ne0$.
Define $p_{n+1},p_{n+2},\dots$ recursively by $p_r = p_{r-1}+...

**9**

votes

**0**answers

97 views

### Factorisation in $\mathbb{N}[X]$?

Do we know an efficient algorithm to factorise in $\mathbb{N}[X]$ ?
One way to do factorisation in $\mathbb{N}[X]$ is to use an algorithm to factorise in $\mathbb{Z}[X]$ and to combine some factor to ...

**5**

votes

**0**answers

128 views

### On factorization algorithms for $\mathcal{O}[x]$

We know that $\mathsf{LLL}$ algorithm provides factorization procedure that runs in poly time for polynomials in $\Bbb Z[x]$ that are primitive.
What other rings $\mathcal{O}$ can we use instead of $\...

**5**

votes

**0**answers

559 views

### Optimal Gear Trains

Suppose you need to slow down a turning motor so that a gear turns at
an angular velocity $\frac{a}{b}$ of that of the motor shaft, where $a$ and
$b$ are natural numbers. For example, this set of ...

**4**

votes

**0**answers

238 views

### Analog of Euler's factoring technique

Is there an analog of Euler's Two Squares factoring theorem over polynomial rings $\Bbb Z[x]$ by considering a version for non-negative polynomials?
Euler's two squares factoring states that numbers ...

**4**

votes

**0**answers

121 views

### More 3-connected cubic graphs with all 2-factors of same cycle type?

The setup is as in this question: Let $G$ be a 3-connected cubic graph. If all 2-factors of $G$ are isomorphic (as graphs), i.e. all have the same partition $\pi$ as cycle type, we'll say that $G$ is ...

**4**

votes

**0**answers

117 views

### Efficiently factorize a KKT system with block diagonal upper corner

I have a system resulting from a quadratic energy minimization with linear equality constraints enforced with Lagrange multipliers which has the form:
\begin{equation}
A =
\left[\begin{array}{c|c}
\...

**4**

votes

**0**answers

601 views

### Least Prime Factor in a sequence of 2n consecutive integers

I was thinking about consecutive integers and I wondered if anyone had done work exploring whether a sequence of $2n$ consecutive integers (i.e. 101,102,103,...,100+2n) always contains at least one ...

**3**

votes

**0**answers

245 views

### Possible $\mathsf{NP}$ complete problem from number theory

A candidate $\mathsf{NP}$ complete variant of factoring was posted in http://cstheory.stackexchange.com/questions/4769/an-np-complete-variant-of-factoring, where decision problem $\text{BOUNDED-...

**3**

votes

**0**answers

112 views

### Coim factorization of a morphism in a complete well-powered category

In a complete, cocomplete and well-powered category with zero object consider the canonical factorization of a morphism $f=k\circ \mathrm{Coim}(f)$. Does cocomplete+complete+well-powered guarantee ...

**3**

votes

**0**answers

156 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 $sr'=r$, ...

**2**

votes

**0**answers

59 views

### When is Coppersmith method polynomial? (Factorization related)

From pari's implementation of Coppersmith method
zncoppersmith(P, N, X, {B=N}): finds all integers $x$ with $|x| \le X$ such that
$\gcd(N, P(x)) \ge B$. $X$ should be smaller than
$$\exp((\...

**2**

votes

**0**answers

110 views

### counting irreducible factors

In How hard is it to compute the number of prime factors of a given integer? a question was asked on computing number of prime factors of an integer.
Suppose we have a polynomial $f(X)\in \Bbb Z[X]$ ...

**2**

votes

**0**answers

134 views

### On reducible polynomials

Let $f(x),g(x)\in\Bbb Z[x]$ with $deg(f)>deg(g)$.
Given an integer $B$, is there any algorithm that runs in $\log^c |B|$ for some fixed $c\in \Bbb R$ to find a $h(x)\in\Bbb Z[x]$ (if one exists) ...

**2**

votes

**0**answers

166 views

### Elements of low multiplicative order and computing square roots modulo composites

In "On computing factors of cyclotomic polynomials", Richard P. Brent
gives the identity
$$ 4 \Phi_n(x) = A_n(x)^2 - (-1)^{(n-1)/2} n B_n(x)^2 \qquad (1) $$
where $n$ is odd squarefree and $A_n,B_n$ ...

**1**

vote

**0**answers

60 views

### On mid common divisor

Given $a,b\in\Bbb N$ of $n$ bits each and $c,d\in\Bbb N$ of $m$ bits each with $\frac{n}\beta<m<n$ with $\beta>1$ is there a better way to decide if $\exists e\in[c,d]\cap\Bbb N$ such that $e|...

**1**

vote

**0**answers

34 views

### Most accurate separation of tensor product of matrices

Given a matrix $A \in \mathbb{R}^{n^m \times n^m}$, how I can find a set of $m$ matrices $\{B_i\}$ such that
$$\arg \min_{\{B_i\, \in\, \mathbb{R}^{n \times n}\}} \|A - B_1 \otimes \ldots \otimes B_m\...

**1**

vote

**0**answers

66 views

### Does Coppersmith's method always finds non-trivial factor of integers of the form $n=a(2^k b+1)$ assuming $1 < a<2^k b +1$ and $b < n^{1/4-0.05}$?

Got an argument and numeric evidence that pari's implementation
of Coppersmith's method finds non trivial factor of integers
of certain form under some assumptions very efficiently.
Three $5000$ bit ...

**1**

vote

**0**answers

76 views

### On variant of integer factorization

In the post on site cstheory.stackexchange on whether a variant of integer factorization
$$\mathsf{}\mbox{ }{\Pi} = \{\langle a, b, n \rangle \;|\; \exists \mbox{ } \mathsf{ an}\mbox{ }\mathsf{ ...

**0**

votes

**0**answers

105 views

### More generalized RSA construction

Is there a way to construct RSA type cryptosystem over general number rings?
Can Number Field Sieve technique be applied here?

**0**

votes

**0**answers

82 views

### Is this factoring algorithm sufficiently efficient for some integers of special kind?

Basically the question is if $m$ is factored over the integers,
can it be relatively efficiently factored over $\mathbb{Z}[\sqrt{n}]$
where $n$ is not factored, but might be of special form?
Suppose ...

**0**

votes

**0**answers

99 views

### What algebras does the hidden subgroup problem for finite abelian groups apply to?

Shor's algorithm is said to solve the hidden subgroup problem for finite Abelian groups, of which factoring and the discrete logarithm problem for integers belong to. Apparently the HSP for finite ...