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

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16
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0answers
272 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
515 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 ...
10
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0answers
101 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
132 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 ...
6
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0answers
194 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 ...
5
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0answers
75 views

Does Pollard rho works for Gaussian integers?

Should I expect that the Pollard rho method ...
5
votes
0answers
182 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 ...
3
votes
0answers
26 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
46 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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0answers
99 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
136 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
votes
0answers
117 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
134 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
votes
0answers
21 views

How Do I Apply the Cantor-Zassenhaus algorithm to $\mathbb{F}_2$?

Recently, I've been trying to implement the Cantor-Zassenhaus algorithm in C++ over $\mathbb{F}_2$. According to this lecture, the algorithm is basically: Input is polynomial $f\in\mathbb{F}_q$ with ...
2
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0answers
29 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
44 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 ...
2
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0answers
171 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
votes
0answers
33 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
60 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
79 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
votes
0answers
63 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
votes
0answers
90 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
votes
0answers
341 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 ...
2
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0answers
85 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
votes
0answers
29 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
119 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
148 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
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0answers
52 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
99 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
203 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
116 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
14 views

Cracking any linear congruential generator

I have a linear congruential generator $X_{n+1} = (aX_n + b) \bmod 2^k $with given arguments and number $Y$. The problem is to find the smallest $i$ that $X_i = Y$ or tell that there is no such $i$. ...
1
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0answers
33 views

ab = X mod Y (X and Y known)

Is there a better way to determine possibilities for $a \mod Y$ and $b \mod Y$ in the following equation: $ab = X \mod Y$ than by brute force? For example if $ab = 5 \mod 6$ then either $a = 1 \mod ...
1
vote
0answers
47 views

Proving that if $p(x)$ divides $f(x)g(x)$ then $p(x)$ divides $f(x)$ or $g(x)$

I need to somehow prove that if $p(x)$ is irreducible and divides $f(x)g(x)$ then $p(x)$ divides $f(x)$ or $g(x)$. I've been given the hints that I should use the theorems: $p(x)$ is irreducible ...
1
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0answers
25 views

factor $\sum_i^4 p_i x^i$

The polynomial $f(x)= \sum_{i=0}^4 p_i x^i $ whose real coefficients are: $$ p_4 = (L^2 + s^2) ^2 \\ p_3= -4 L^2 (L^2 + s^2)\\ p_2 = 6 L^4 + 2 L^2s^2 \\ ...
1
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0answers
32 views

Factorization of polynomial $f(x,y)$

The motivation is solving the following equations: $$ f(x,y)=0, x=L-kL, y=ks $$ $k$ is the variable, $L$ and $s$ are constants. The plan is: First, to factorize the polynomial as $(a_1x+b_1y+c_1) ...
1
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0answers
11 views

Complex filter factorizations with invariant points

Based on this question, using the same $z_0$: $$z_0 = e^{2\pi i / 8}$$ if we modify the sequence from previous question to look like this ($*$ denotes discrete convolution): $$\left(z_0^{[-2k,3k]} * ...
1
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0answers
17 views

Complex filter factorizations - continued

Continuing from this rather silly trivial question factoring real valued filters into shorter complex ones, hoping this won't be as trivial. If we modify it a bit: $$z_0 = e^{2\pi i / 8}$$ and ...
1
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0answers
27 views

Factoring Algorithm Using Multilinear Forms

I was wondering if anyone could identify for me a known name for this algorithm, I would appreciate it. I will give an example. Let $N=961$. Then write $(6x+1)(6y+1)=6(160)+1$ Where $0<x\le y$ ...
1
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0answers
44 views

Reason for the method of factorization of cyclic expressions.

For example, I am given that factorize: $$a^2b+a^2c+ab^2+2abc+ac^2+b^2c+bc^2$$ So by the traditional method, we take the powers of $a$ $$=a^2(b+c)+a(b+c)^2+bc(b+c)$$ $$=(b+c)(a^2+ab+ac+bc)$$ ...
1
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0answers
35 views

Factor out factorial from expression

I have the expression $(k+1)! - 1 + (k+1)(k+1)!$ How do I factor out $(k+1)!$ to achieve the result: $[(k+1)!(k+2)] - 1$? I for the life of me cannot figure this out. Thanks!
1
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0answers
15 views

How can I factorize $|z_1|^q z_1 - |z_2|^q z_2$?

Let $1\le q\le 2$. I would like to know that how I can factorize the following: $$ |z_1|^q z_1 - |z_2|^q z_2 = \\(z_1 - z_2) (\text{ a function of } z_1, z_2, \overline{z_1}, \overline{z_2}, q ) + ...
1
vote
0answers
86 views

Finding roots and factors of multivariate polynomials

I know that in order to factor a one dimensional polynomial one can find the roots with some method, for instance a numerical newton method. Then one can systematically divide with $(variable-root)$ ...
1
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0answers
39 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 $n$ ...
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0answers
39 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
82 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
vote
0answers
37 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
vote
0answers
45 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 ...