For questions about finding factors of e.g. integers or polynomials

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14
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0answers
229 views

Irreducibility of $~\frac{x^{6k+2}-x+1}{x^2-x+1}~$ over $\mathbb Q[x]$

The Artin—Schreier polynomial $~x^n-x+1~$ is always irreducible over $\mathbb Q[x]$, unless $n=6k+2$, in which case it seems to have only two factors, one of which is always $x^2-x+1$. The ...
12
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0answers
397 views

How does a Lehmer Sieve work?

http://en.wikipedia.org/wiki/Lehmer_sieve Apparently a Lehmer Sieve was a mechanical device that used chains and pulleys to factor numbers and solve diophantine equations. It once was able to factor ...
9
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0answers
87 views

Factorize: $a^3(b+c)+b^3(a+c)+c^3(a+b)$.

Factorize: $a^3(b+c)+b^3(a+c)+c^3(a+b)$. I found this question on a high school textbook but it seems impossible to be further factorized. The best I can get is: $a^3(b+c)+b^3(a+c)+c^3(a+b) = ...
8
votes
0answers
122 views

Is there an easy way to factor polynomials with two variables?

On a recent precal test, I saw a question involving the following expression: $$(x+1)^2-y^2$$ Which factored out into: $$(x+y+1)(x-y+1)$$ This wasn't very hard, considering that it was already ...
5
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0answers
178 views

Fastest Primality test using $N-1$ Factorization?

If $N-1$ could be factored easily with several small prime factors, then what is the fastest way to check $N$ for primality? Updated I'm aware of Pocklington primility test which is not good for ...
5
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0answers
218 views

Carmichael number factoring

The task I'm faced with is to implement a poly-time algorithm that finds a nontrivial factor of a Carmichael number. Many resources on the web state that this is easy, however without further ...
4
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133 views

Reducing multivariate rational fractions to lowest terms

I wish to simplify multivariate rational fractions to a canonical form. Thanks to some very helpful mathematically inclined people who verified that my understanding of Wikipedia was correct, I'm now ...
3
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0answers
21 views

What was Gauss' 2nd Factorization Method?

Reading Jean-Luc Chabert's A History of Algorithms, I learned that Gauss, prompted by the poor state-of-the-art, designed two distinct methods for fast integer factorization. Chabert's book discusses ...
3
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0answers
22 views

For any field $k$ and $n>0$: $X^n-a\text{ reducible }\ \Leftrightarrow\ a=b^p\ \vee\ a=-4b^4$.

I'm stuck on the following exercise: Let $k$ a field and $n$ a positive integer. Prove that $X^n-a\in k[X]$ is reducible if and only if there exist a prime $p$ dividing $n$ and a $b\in k$ such ...
3
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91 views

proof that $\frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1}$ is irreducible

I am reading the group theory text of Eugene Dickson. Theorem 33 shows this polynomial is irreducible $$ \frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1} \in \mathbb{Z}[x]$$ He shows this polynomial ...
3
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0answers
122 views

Probability distribution of the (size of the) smallest prime factor

Related question: Expected smallest prime factor Background: Given a toolbox of factorization algorithms (like trial division, ECM, quadratic sieve, GNFS) and a set of large composite numbers, I'm ...
3
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0answers
106 views

Symmetry in the Sum of Factors

The factors of 72 are 1,2,3,4,6,8,9,12,18,24,36,72. The factors can be organized as such $$m=1,2,3,4,6,8$$ $$n=72,36,24,18,12,9$$ Knowing the symmetric properties and that the 12 factors exist. Pair ...
3
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0answers
131 views

Why does Lenstra ECM work?

I came across Lenstra ECM algorithm and I wonder why it works. Please refer for simplicity to Wikipedia section Why does the algorithm work I NOT a math expert but I understood first part well enough ...
2
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0answers
37 views

Notation for indexing the factorizations of a number?

Background Given any $n \in \mathbb{N}$, the ordered factorization count of $n$ can be computed and is traditionally written $H(n)$. This is, essentially, the number of unique decompositions of $n$ ...
2
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26 views

Factorization by multiplying and representation as difference of two squares

Definition 1.$$R: \mathbb N \to \mathbb N: \ R(n) = \lceil\sqrt{n}\rceil^2-n.$$ This is the distance from $n$ to the smallest square greater or equal to $n$. Definition 2. Let $a$ be as positive ...
2
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0answers
75 views

A quick way to determine number of ring homomorphism

Find the number of ring homomorphism from $\mathbb Z_2\times \mathbb Z_2$ to $\mathbb Z_4$. I know that checking idempotent elements the number of ring homomorphisms is $1$. Again the number of ring ...
2
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0answers
29 views

About factoring trinomials over $\mathbb{Z}$

We were taught in school an algorithm to factor a trinomial of the form $$x^2\pm bx\pm c$$ with $b,c\in \mathbb{Z}^+$. Assuming the best scenario (that the polynomial has both roots in ...
2
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0answers
54 views

Multiplication Sieving

Background I made another improvement to Fermat's sieve factorization, by merging the sieves groups of $4,3,5,7,11,13,17$ in two one big group. This method allows me to reduce the values that I need ...
2
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0answers
67 views

Does this approach for factorizing RSA numbers help in any way?

I was thinking about why factorizing RSA numbers is so hard. When humans perform any kind of maths manually, they often employ various 'tricks' that get them closer to the answer. Some are based on ...
2
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0answers
78 views

Show that $x^4+n(2-n)x^2+n^2$ reducible over Q for any natural number n.

Show that $$x^4+n(2-n)x^2+n^2$$ reducible over Q for any natural number n. Here is what I did $$x^4+n(2-n)x^2+n^2=x^4+2nx^2-n^2x^2+n^2=x^4+2nx^2+n^2-n^2x^2$$ $$=(x^2+n)^2-(nx)^2$$ ...
2
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0answers
55 views

factor theorem for multivariables

My understanding of the remainder theorem for one variable is that for $$f(x)=(x-a)q(x)+r(x)$$$\qquad$ if $x=a\implies f(a)=0\times q(a)+r(a)$ so $f(a)=r(a)$ Is this correct for a multivariate ...
2
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0answers
89 views

Humankind knows the prime factorization of the first how many consecutive integers?

I am only looking for an approximation. I'm guessing the answer must be somewhere between $10^{20}$ and $10^{50}$. . Edit: Okay so my first initial estimation was pretty poor... I should have ...
2
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0answers
83 views

What is the line of thinking to get $[(n^2+3n+1)^2-5n(n+1)^2]$ from $(n^4+n^3+n^2+n+1)$?

There is this one little tiny step along the working that I don't quite understand, but I think it is better if I write the whole problem and solution for clarity. Problem: Factorise $5^{1995}-1$ ...
2
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0answers
27 views

Decomposability of a function into two

This is question on the existence of a decomposability of a function $f(v)$ into multiplicative factors $\lambda(v)$ and $g(v)$. The question is non-trivial since the functions $\lambda(v)$ and $g(v)$ ...
2
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0answers
114 views

Quadratic Diophantine Equations in Polynomial Time

Considering the problem of finding lattice points $(x_1, x_2 ... x_n)$ that satisfy a quadratic law: $F(x_1, x_2... x_n) = 0$ such that $F(x_1, x_2... x_n)$ is a second degree polynomial It is ...
2
votes
0answers
121 views

Prime factors of a random number

Let $r$ be a uniformly random integer between $1$ and $N$, for some large enough $N$ (i.e., I'm only interested in the asymptotics). What is the expected largest prime factor of $r$? Is there a good ...
2
votes
0answers
51 views

Different elliptic curves over given $\mathbb{F}_q$ can have different orders?

As the title says: For a given $q \in \mathbb{N}$, in $\mathbb{F}_q$, is it true that different curves on it can have different group orders? I assume yes. Let $q:=5$. Wikipedia gives $9$ points for ...
2
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0answers
93 views

Understanding of Pollard rho factorization

I am trying to better understand the ideas and intuition behind the Pollard Rho factorization algorithms. Given an $x_0$ and an irreducibe polynomial we can create a sequence from the recursive ...
2
votes
0answers
194 views

How many co-primes are there for a big integer N over a specified interval?

How many co-primes are there for a big integer $N$ over a specified interval ? bounds of $N$ are $[1,10^9]$ and the interval is $[a,b]$ where ($1\leq a\leq b \leq 10 ^{15}$) and there are $100$ ...
2
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0answers
110 views

Factors of a polynomial in several variables

Fix an embedding of $\overline{\mathbb{Q}}$ into $\mathbb{C}$. Suppose you have a polynomial in several variables, with algebraic coefficients: $P\in \overline{\mathbb{Q}}[z_1, \ldots, z_n]$. Also ...
1
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0answers
34 views

Factorizing a Polynomial over the Integers

What are the most efficient algorithms to factorize a polynomial over integers, knowing that it has only integer roots? I googled around a lot, but most of the work seems to be around Finite fields. ...
1
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0answers
31 views

Factoring quaternion into three parts

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 ...
1
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0answers
71 views

Factorisation of large polynomials and Galois theory

As I understand it, one of the consequences of Galois theory is that there is no way of expressing the solutions to a general polynomial of degree 5 or higher in terms of radicals. Would a theory that ...
1
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0answers
37 views

Solving “finding value” using factorisation

Even my teacher could not do this.This is a question extracted from the mathematics challenge board created by my friend in school.This is the question Given $x-y=3$,find the value of ...
1
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0answers
27 views

How is pollard rho different from normal factorization?

As far as I understand, pollard rho factorization generates random sequence of numbers, say x1, x2, x3 ... and then checks if x(i) - x(i-1) divides the number. If it does then it is a factor. How is ...
1
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0answers
19 views

Splitting field of a cubic polynomial understanding

The cubic polynomial $f(x) = x^3+px+q\in K[x]$ has 3 roots $a_1,a_2,a_3\in \mathbb C$ Hence, the splitting field extension $L=K(a_1,a_2,a_3)$ $\delta=(a_1-a_2)(a_1-a_3)(a_2-a_3)\in L$ since ...
1
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0answers
26 views

Factorising Complex Polynomial with Complex Coefficients

I have tried to factorise the polynomial in question 19 by using the factor theorem to find other factors, however this has been unsuccessful thus far. Seeing as the conjugate root theorem does not ...
1
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0answers
36 views

When looking at the mod as binary value

Look at the next value: $$617*947 = 584299$$ 617, 947 are prime values. I want to see what are all the possible solutions for the next equation, for $k=4$: $$(a\mod k)(b\mod k) = 584299\mod k$$ ...
1
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0answers
37 views

When $\frac{-1436+y\pm \sqrt{(1436-y)^2 - 4\cdot480547}}{-2}$ is integer

While working on my algorithm I came to this problem: $$x_{1,2} = \frac{-b-a+y \pm \sqrt{(b + a-y)^2 - 4c}}{-2}$$ $$x_1 = \frac{c}{x_2}$$ $a,b,c$ are positive known integers $y$ is positive integer ...
1
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0answers
23 views

Should this be rephrased into saying no common factors but 1?

This is from Discrete Mathematics and its Applications For the phrase "a and b have no common factors" , does that actually mean a and b have no common factors other than 1? I feel like this would ...
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0answers
56 views

Findind 3 factors for a integer number

As a background I'll explain what I'm trying to achieve and where's from. In Person of Interest, a TV series, one of the characters gets a phone number in the form of area code and phone number, like ...
1
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0answers
93 views

Factoring a polynomial over $\mathbb F_{2^8}$

How do you find the factors of $x^4+x+1$ in $GF(2^8)$ in terms of polynomials? Let me explain, We have primitive irreducible polynomial $p(x)=x^2+x+1$ in $GF(2^2)$ which has root $\alpha^2+\alpha$ in ...
1
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0answers
38 views

Numbers with special factorisation

We know that any natural number $n$ can be decomposed as $p_1^{k_1}p_2^{k_2}...p_n^{k_n}$. I am looking for numbers which have $k_1=k_2=k_3=....=k_n=1$ i.e. given a number n, identify if it has all ...
1
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0answers
74 views

Binary Polynomial Factoring

I just need confirmation that I've done my math right. If $a(x) = x^4 + x^3 + x + 1$ and $b(x) = x^2 + x + 1$ are binary polynomials, find binary polynomials s(x) and r(x) such that $x^4 + x^3 + x + ...
1
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0answers
24 views

Finding coefficients of the min polynomial of an $n\times n$

Given an $n\times n$ matrix, for ease assume this matrix is over the $F_m$. What we know about min poly is the the non-zero components of the min polynomial for this case, ie if there is $x^2$, or ...
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0answers
26 views

Factorisation algorithm for polynomials in several variables over $\mathbb{Z}$.

What algorithm is used by a CAS to decide how to factor a polynomial in several variables over $\mathbb Z$?
1
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0answers
201 views

Proof for maximal ideals in $\mathbb{Z}[x]$

I have been trying to prove the following theorem: Every maximal ideal in $\mathbb{Z}[x]$ has the form $(p, f(x))$ where $p$ is prime integer and $f$ is primitive integer polynomial that is ...
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0answers
17 views

By what factor do winning chances increase based on total value?

Say I am entering 24/7 in endless sweepstakes, contests, giveaways, drawings, etc. Assuming each one I enter has no less than 1 in 1,000 chances, but no more than 1 in 1 million (and I enter at least ...
1
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0answers
49 views

On number of different factorizations over integers of a number field

Let $K$ be a finite field extension of the rational numbers and let $\mathcal{O}_K$ denote its ring of integers. If a rational integer $n$ factors into two distinct ways into irreducible elements in ...
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0answers
77 views

How are the general skills for complete factorization of arbitrary homogeneous order polynomials in $\mathbb{C}$?

How are the general skills for complete factorization of arbitrary homogeneous order polynomials in $\mathbb{C}$ ? For example: $1.$ $a^2+b^2+c^2$ $2.$ $a^2+b^2-c^2$ $3.$ $a^2+b^2+c^2+d^2$ $4.$ ...