The factorization tag has no usage guidance.

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### 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 ...

**18**

votes

**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 ...

**16**

votes

**11**answers

4k views

### Fastest way to factor integers < 2^60

I've been running a search for Mordell curves of rank >=8 for about 12 months and have identified approximately 280,000 curves in our archivable range, amongst many millions that aren't.
For this ...

**16**

votes

**1**answer

3k views

### Evidence for integer factorization is in $P$

Peter Sarnak believes that integer factorization is in $P$. It is a well-known open problem in TCS to identify the real complexity class of integer factorization. Take a look at this link for Peter ...

**16**

votes

**1**answer

1k views

### Factoring Integers using Complex Integrals

Suppose $n$ is an integer and we wish to factor it. As a special case we have $n = pq$ with $p,q$ distinct primes. The problem: factoring $n$ via complex analysis tools
Background
I have been ...

**14**

votes

**6**answers

2k 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 ...

**14**

votes

**2**answers

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 ...

**14**

votes

**1**answer

459 views

### Is the ring of quaternionic polynomials factorial?

Denote by $\mathbb{H}[x_1,\dots,x_n]$ the ring of polynomials in $n$ variables with quaternionic coefficients, where the variables commute with each other and with the coefficients. Two polynomials ...

**12**

votes

**2**answers

637 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 ...

**11**

votes

**2**answers

815 views

### Saying things rapidly about integer factorisations

Let $N$ be a positive integer. Thanks to the Miller-Rabin test and the work of Agrawal, Kayal and Saxena, these days people have much much faster algorithms for testing whether $N$ is prime or ...

**11**

votes

**3**answers

1k views

### What is known about the polynomial factorization of power series?

Some power series factorize; $1+\sum_{n=1}^\infty x^n=\prod_{n=1}^\infty (1+x^{2^n})$ and $1+\sum_{n=1}^\infty x^{2n}/(2n+1)!=\prod_{x=1}^\infty (1+x^2/n^2\pi^2)$ for example; while others do ...

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**0**answers

324 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 = ...

**10**

votes

**2**answers

483 views

### Why going to number fields in number field sieve help beat quadratic sieve?

To factor an $n$ bit integer number field sieve roughly takes $$e^{c{(\ln\ln n)^{\frac23}}({\ln n})^{\frac13}}$$ time while quadratic sieve takes $$e^{c{(\ln\ln n)^{\frac12}}({\ln n})^{\frac12}}$$ ...

**9**

votes

**5**answers

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### Polynomials all of whose roots are rational

I have two questions about the class of integer-coefficient polynomials all of whose roots are rational.
I asked this at MSE, but it attracted little interest (perhaps because it is not interesting!)
...

**9**

votes

**3**answers

2k 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 ...

**8**

votes

**1**answer

544 views

### Is this pleasing polynomial irreducible?

Let:
$f(x)=x^n+2x^{n-1}+3x^{n-2}+4x^{n-3}+\ldots + (n-1)x^2+nx+(n+1)$.
Is $f(x)$ irreducible?
In light of the answers to this question, I now know that this is true when $n+1$ is prime. What about ...

**8**

votes

**2**answers

680 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 ...

**8**

votes

**2**answers

5k 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 ...

**7**

votes

**4**answers

2k 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 ...

**7**

votes

**2**answers

466 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 ...

**7**

votes

**4**answers

574 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 ...

**7**

votes

**2**answers

328 views

### Theorems of the Galois groups of quintics appears not to work for the ${F}_{20}$ group determination

I am computing the Galois groups of quintics using the theorems from Ryan Kavanagh paper "On Irreducible Rational Quintics" using the decic resolvent ${P}_{10} \left({x}\right) = \prod\limits_{1 \le i ...

**7**

votes

**3**answers

666 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 ...

**7**

votes

**1**answer

512 views

### Divisibility and factorization in rings that are not integral domains

In my course notes for an undergraduate course "Algebra I", I wrote at the point when I'm introducing the notion of divisibility in rings (in a section on unique factorization):
We want to study ...

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votes

**2**answers

634 views

### Factoring some integer in the given interval

I'm posting this question here (rather than on CSTheory) since it seems to require much more knowledge about number theory than algorithms.
Let N be a positive integer. Is there an efficient (i.e. ...

**7**

votes

**2**answers

260 views

### A Different 2-factor in a graph

We know that a k-factor of G is a k-regular spanning subgraph of G. And if G is 4-regular (or 2k-regular), it can be partitioned into 2 (k) edge-disjoint 2-factors (Petersen 1891).
My question is in ...

**7**

votes

**1**answer

123 views

### Intuition for density comonad in relation to lifting problems

In Emily Riehl's Categorical Homotopy Theory, there is a section on Garner's Small Object Argument which I'm trying and failing to understand. Originally I followed most of Garner's paper, using the ...

**7**

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**2**answers

287 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 ...

**7**

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**3**answers

813 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 ...

**7**

votes

**1**answer

218 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

**2**answers

365 views

### Parametrization of positive semidefinite matrices

We know that a real, symmetric, positive definite matrix $A$ of size $n\times n$ can be parametrized by a vector $\theta$ of $\frac{n(n+1)}{2}$ parameters thanks to the Cholesky decomposition:
$$
A = ...

**6**

votes

**1**answer

446 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) ...

**6**

votes

**3**answers

2k views

### Reduction from factoring to solving Pell equation

The paper Polynomial-Time Quantum Algorithms for Pell's Equation and the Principal Ideal Problem claims
There are reductions from factoring to solving Pell’s equation, and from solving Pell’s
...

**6**

votes

**1**answer

171 views

### Shortest vector problem over polynomials

In shortest vector problem, given a lattice in $\Bbb Z^n$, we seek the shortest non-zero vector in the lattice. This problem is computationally difficult.
Is there a polynomial analog of this problem ...

**5**

votes

**1**answer

164 views

### Dynamics of the distribution of prime factorization types in increasing intervals

I've tagged this as reference request as surely this question must be very well investigated, I just don't know how to look for it. Most likely the perfect answer will be in form of a keyword for ...

**5**

votes

**1**answer

209 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 ...

**5**

votes

**2**answers

163 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 ...

**5**

votes

**0**answers

123 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

**1**answer

273 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 ...

**5**

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**0**answers

547 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 ...

**5**

votes

**0**answers

561 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 ...

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votes

**3**answers

486 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 ...

**4**

votes

**2**answers

1k 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 ...

**4**

votes

**2**answers

328 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 ...

**4**

votes

**1**answer

1k 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 ...

**4**

votes

**3**answers

417 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 ...

**4**

votes

**1**answer

458 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 ...

**4**

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**0**answers

235 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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**0**answers

119 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

114 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}
...