The factorization tag has no usage guidance.

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### Decomposition of orthogonal matrix into 2 orthogonal matrices [on hold]

Is there anyway to find a decomposition of orthogonal matrix $A$ into 2 orthogonal matrices $P$ and $Q$ such that $A = PQ^T$?

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65 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 ...

**2**

votes

**1**answer

196 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 ...

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215 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 ...

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**1**answer

176 views

### Find all possible rational pairs of a parametric sextic and all cases where it is reducible for either parameter

Find all possible rational solutions pairs $(z,a)$ of the equation
$a^6 + a^4 (-18368 + 9184 z - 2912 z^2) + a^2 (61702144 - 61702144 z + 36814848 z^2 - 10694656 z^3 + 1748992 z^4) - z^2 ...

**3**

votes

**1**answer

86 views

### $P_3$-factors for 3-regular, 3-connected cubic graphs

Suppose that $G=(V,E)$ is a simple graph.
We know if $G$ is 3-regular, 3-connected and $|V|=4k$ for some $k\in \mathbb{N}$, then $G$ has a $P_4$-factor.
Question. Let $G=(V,E)$ be 3-regular, ...

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**2**answers

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 ...

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45 views

### Factoring quaternion into three parts [closed]

I am faced with a problem where I have a quaternion which represents rotation and three arbitrary axis about which I can make rotations, thus three unit vectors.
What I would like to know is angles ...

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**4**answers

1k views

### How did Cole factor $2^{67}-1$ in 1903

I just heard a This American Life episode which recounted the famous anecdote about Frank Nelson Cole factoring $N:=2^{67}-1$ as $193707721 \times 761838257287$. There doesn't seem to be a historical ...

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222 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 ...

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416 views

### Irreducible/prime/indivisible elements

in what follows all the rings are commutative, nontrivial, with unit.
Recall the following definitions:
1) $\pi\in A$ is prime if $(\pi)$ is a nonzero prime ideal
2) $\pi\in A$ is irreducible if ...

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**1**answer

131 views

### On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$, Part II

I posted this question on MSE two days ago, but did not receive any responses. I have cross-posted it on MO, hoping it gets more attention here and that it is appropriate for this site.
A positive ...

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**2**answers

293 views

### Find all possible rational values of a parametric quartic such that it is reducible

Description: Given the following parametric quartic polynomial
$y^4 - 28 z y^3 - 14 (656 - 328 z + 83 z^2) y^2 +
4 z (-20464 + 10232 z + 3409 z^2) y +
91 (62208 - 62208 z + 41504 z^2 - 12976 z^3 ...

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209 views

### Algorithm for fast factorization of polynomial over $\mathbb Z$ or over $\mathbb F_p$

I want to fast decompose polynomial over ring of integers (original polynomial has integer coefficients and all of factors have integer coefficients) and also over ring of integers modulo prime ...

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**1**answer

220 views

### Find all possible rational values of the parameter of a parametric cubic such that it is reducible

Description: Given the following parametric cubic polynomials ${E}^{3}
- 15\, {\beta}_{\pm}\, {E}^{2} - 3 \left({71\, {\beta}_{\pm}^{2} + 352\, {\beta}_{\mp}}\right) E
+ 135\, {\beta}_{\pm} ...

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439 views

### $2^n$-1 consisting only of small factors

I've checked the factorization of $2^N - 1$ up through N = 120 for the largest prime factor, and it looks like the largest value of N where $2^N-1$ has a largest prime factor under 2500 is N = 60 ...

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607 views

### Factorization when a factor is partially known

Let's say that I have a very large number of the order ($10^{250+}$) which is composite. I have been given one of its factor partially to a significant amount of digits (say 75+). Then, how can I ...

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**1**answer

70 views

### On a reciprocal of Ostrowski theorem on Newton polytopes and factorization

$\newcommand\KK{\mathbb{K}}$Let $\KK$ be any field and $f\in\KK[x_1,\dotsc,x_n]$ be a polynomial. Its support $S_f$ is the set $\{(e_1,\dotsc, e_n) : x_1^{e_1}\dotsb x_n^{e_n}$ has a nonzero ...

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**1**answer

152 views

### Irreducibility of Faulhaber-like Polynomials over $\mathbb Q[x]$

Motivation: Inspired by the famous Faulhaber polynomials $F_k(N)=\displaystyle\sum_{n=0}^Nn^k,$ I decided to study their alternating versions, $\Phi_k(M)=\displaystyle\sum_{n=0}^M(-1)^nn^k$.
For ...

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100 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) ...

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2k views

### Finding the greatest (smallest) factor of a number smaller (greater) than another number

Instead of iterating through all the possible numbers, is there a better way to find the greatest factor of a number $n$, such that it is less than $m$ ($m$ < $n$). Similarly how does one find the ...

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313 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 = ...

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**1**answer

2k views

### Difficulty of factoring a Gaussian integer (compared to factoring its norm)

Given a Gaussian integer $G=a+ib$, with $gcd(a,b)=1$, a well-known strategy for factoring $G$ is to first compute its norm $N(G)=a^2+b^2$, factor the norm and finally recover the correct generator ...

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101 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 $f(X)\in \Bbb Z[X]$ ...

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**1**answer

117 views

### Which nonnegative matrices have exact nonnegative matrix factors of smaller dimensionality?

The nonnegative matrix
$V = \left( \begin{array}{cc}
1 & 1 \\
1 & 1 \end{array} \right)$
has nonnegative matrix factors $W = \left( \begin{array}{c} 1 \\ 1 \end{array} \right)$ and $H = ...

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105 views

### Puzzling CAS-detected factorization by cyclotomic polynomials [closed]

If one factorizes by CAS the expression $$x^{\frac{m(m+1)}{2}}\prod_{k=1}^m(x^k+(\frac{1}{x})^k)$$ a puzzling perfect factorization seems to be possible for all natural values of $m$.
E.g. for ...

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99 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 ...

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255 views

### Cubic graphs whose 2-factors all have the same cycle type

Let $G$ be a bridgeless cubic graph. I am interested in such graphs where all 2-factors are isomorphic (as graphs), i.e. have the same partition as cycle type. We'll say that this partition is ...

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615 views

### runs of consecutive non squarefree integers

This question gained no attention at Math SE.
Call a sequence of $k$ consecutive naturals squary if each one of them is divided by a square > 1. The Chinese Remainder theorem trivially guarantees us ...

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358 views

### Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$

I’m facing the problem of factoring polynomials of type $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ and the degrees of factors seem to be quite special. For example, according to my experimental results ...

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403 views

### Distinct primitive factorizations over integers of number fields

I am curious about the following.
Let $K$ be a number field. For any $a \in \mathcal{O}_K$ in its ring of integers, let $N(a)$ be zero if there exist elements $b, c \in \mathcal{O}_K \setminus ...

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95 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}
...

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80 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 ...

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**1**answer

175 views

### Given an arbitrary composite odd integer $N$, find two integers $P$ and $Q$ such that $P-Q \neq 1$ and $N=P^2-Q^2$ [closed]

Given an arbitrary composite odd integer $N$, find two integers $P$ and $Q$ such that:
$P-Q \neq 1$ and $N=P^2-Q^2$
I am assuming that the best known solution to this problem runs at $O(2^{|N|})$.
...

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**1**answer

348 views

### Proving conditions on $(r+s)^2 \mid (4r^4+1)$, related to Pell oblongs

While working on another problem (Solving the quartic equation $r^4 + 4r^3s - 6r^2s^2 - 4rs^3 + s^4 = 1$), I came across a question which seems to be of [semi-] independent interest.
Conjecture. If ...

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**1**answer

202 views

### Large gaps between consecutive irreducible polynomials with small heights

For a prime gap of length at least $n$, a trivial upper bound for its first occurrence is $N=n!$ or $N=lcm(2,\dots,n)$. A bit better is $N=p_1\cdots p_n$ where $p_k$ is the $k$th prime, as then ...

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**1**answer

133 views

### Aurifeuillean factorization with number fields

Basically the question is if number fields can be used
in Aurifeuillean factorization.
Probably this is easy and the answer is "no".
Let $f,g \in \mathbb{Z}[x], a \in \mathbb{N}$.
Let $f(x)$ and ...

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**4**answers

520 views

### Are there any fast algorithms for factoring integers that don't work by searching for smooth numbers?

All of the fast algorithms that I have seen which factor integers work by searching for smooth numbers. Are there any fast algorithms for factoring integers that don't work by searching for smooth ...

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**1**answer

783 views

### The number of distinct prime factors of $n\in\mathbb N$

Let $\omega(n)$ be the number of distinct prime factors of a natural number $n$.
Note that $\omega(n)=0\iff n=1$, and that $\omega(24)=\omega(2^3\cdot 3^1)=2\ (\not = 4)$.
(For more details, you ...

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824 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 ...

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**2**answers

402 views

### Number of ways to write an integer as a product of irreducibles

Is there any way to tell the number of distinct ways to factor $a\in\mathcal{O}_k$ (up to units, of course) when $k$ is not a PID? A simple investigation in $\mathbb{Q}(\sqrt{-5})$ with integer ring ...

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103 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 ...

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**1**answer

403 views

### Using the decomposition $641 = 5^4 + 2^4$ to factor $F_5$

The question in the title arises from a problem in Stewart's "Galois Theory, Third Edition" (and possibly elsewhere) which has been bugging me for a few days since reading it:
Problem 19.5 (p. 224) ...

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796 views

### $\omega(p^n - 1)$ as $n \rightarrow \infty$

Although I am also interested in the number of distinct prime factors (not counting
multiplicity), today I use $\omega(m)$ to denote the number of (positive) prime
factors (with multiplicity) of the ...

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**3**answers

533 views

### For any prime $p$, is there $C$ such that if $x\ge C$, then all but one integer among $x+1, x+2, \dots, x+p$ has Greatest Prime Factor $> p$

I apologize if this is a naive question about greatest prime factors (gpf). I was thinking about the sequence of integers where $\mathrm{gpf}(x) \le p$ where $p$ is any prime.
Clearly, as $x$ ...

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158 views

### Collision polynomials

Consider $P_n(x)$ polynomials defined through the recurrence relations
$$P_n(x)=2(1-x)P_{n-1}(x)-(1+x)^2P_{n-2}(x),$$ with $P_0(x)=1$ and $P_1(x)=1-3x$.
In fact, the explicit solution of these ...

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151 views

### Significance behind splitting of (x+y)^7-(x^7+y^7)? [closed]

A quick check via third roots of unity establishes$$(x+y)^7-(x^7+y^7) = 7xy(x+y)(x^2+xy+y^2)^2$$
Some questions:
What is the significance of this factorization?
Does it have any connection to the ...

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158 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 $A_n,B_n$ ...

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**6**answers

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 ...

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**2**answers

254 views

### Generating 3SAT circuit for Integer factorization example

I read somewhere that 3SAT can be used to solve Integer Factorization.
If that is true, could someone teach me a simple example of generating the 3SAT by using a small number? Let's say you are given ...