The factorization tag has no wiki summary.

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Find the smallest number (within a range) that either strictly divides or is a strict multiple of a given set of numbers [closed]

Given a set of positive numbers. I need to find a number (not in the set) from a given range such that it is either a multiple or a divisor of the numbers of the set. How should I go about it?
...

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### 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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### 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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### 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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### 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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### 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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### 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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### $\omega(p^n - 1)$ 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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### 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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### 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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### 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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### 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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### 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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### 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 ...

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### How can an integer be factorized as n*m so that n^m has the highest value. [closed]

Given an integer d, is there a way of finding max(n^m), where d=n*m, except brute-forcing it ?
n and m don't have to be primes.
For example:
The number 12 can be factorized into two factors as: ...

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### 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 ...

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### Can infinite polynomials be expressed as a product of its linear factors?

Background:
In the 1700s, Euler solved the Basel Problem, which was to solve $\sum_{n=1}^\infty\frac{1}{n^2}$ in closed-form. Euler showed that it was equal to $\frac{\pi^2}{6}$ by first expressing ...

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### Factorization of antisymmetric bounded holomorphic functions

A basic principle in complex function theory is that one can split off zeros of holomorphic functions in a similar way as for polynomials: If $f$ is holomorphic near $0$ and $f(0) = 0$, then $f(z) = ...

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### 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 ...

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### 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 ...

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### Groups of Rational Points on Gaussian Circles

Let a gaussian circle $C_R$ be any circle defined by the equation:
$$x^2+y^2 = R, (x,y) \in \mathbb{R}^2$$, where $R$ is the norm of a gaussian integer ($R=a^2+b^2, (a,b) \in \mathbb{Z}^2$). IF $R$ ...

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### Algorithm for detecting prime powers

While reading Peter Shor's paper Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, I came across the following quote:
"This scheme will thus work as ...

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### 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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### 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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### find the minimum difference between the factors of a number

Given a number c, what is the smartest way to find |x - y| such that x * y =c and |x - y| is minimum

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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!)
...

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### Factoring a certain quartic mod primes

Let $h(x)=x^4+12x^3+14x^2-12x+1$, and let $p>5$ be a prime.
I want to show $h(x)$ factors into 2 quadratics mod $p$ if $p \equiv 9,11$ mod 20, while
$h(x)$ factors mod $p$ into 4 linear factors if ...

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### 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 ...

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### 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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### Finding a divisor of a number under a constraint

Given 2 positive integers $n, l$ with $ l \leq n$, I am looking for a way to find the largest divisor $d$ of $n$, such as $d \leq l$.
Assume $n$ has too many divisors for an exhaustive search.
Thanks ...

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### Are these polynomials irreducible over ring Z of integers ?

Is it true that polynomials of the form :
$ f(x)= x^n+x^{n-1}+...+x^{k+1}+ax^k+ax^{k-1}+...a$
where $gcd(n+1,k+1)=1$ and $ a\in \mathbb{Z^{+}} $
are irreducible over ring $\mathbb{Z} $ of ...

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### 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 ...

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### 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
...

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### 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 ...

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### Factoring and Index Calculus and duality between DL and factoring via compuational problems made easy through them

If factoring is in $P$ (with a blazing fast polynomial time in $P$), would it affect the index calculus algorithm used for Discrete Log calculation in any serious way?
Other connections
$1.)$ ...

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### 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 ...

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### 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. ...

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### 10 factors for x^2 coefficient in quadratic sieve?

I wrote a quadratic sieve and I tried plugging in all the same parameters as the wikipedia article says msieve uses: http://en.wikipedia.org/wiki/Quadratic_sieve#Parameters_from_realistic_example
It ...

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### 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, ...

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### 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 ...

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### How many minimal pair-wise coverings can there be?

Suppose I have a set of finite sets: $X = \{V_1, V_2, V_3\}$ where each set called $V_i$ in $X$ contains a number of symbols (i.e. $V_1 = \{a,b,c\}$). $Z$ contains all of the Cartesian products of ...