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

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9
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0answers
239 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 ...
8
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0answers
103 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
35 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 ...
5
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0answers
165 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
191 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 ...
3
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0answers
92 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
106 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
80 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
84 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 ...
2
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0answers
77 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
23 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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101 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
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0answers
49 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
171 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
97 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

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
36 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
21 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
24 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 $\mathbb Z$?
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96 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
13 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 ...
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0answers
40 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
18 views

Which factors determine whether a set of variables are suitable for factor analysis?

Which factors determine whether a set of variables are suitable for factor analysis? I am looking as much for an explanation of the question as a tentative answer to it. So grateful for any help on ...
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0answers
69 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: $1.$ $a^2+b^2+c^2$ $2.$ $a^2+b^2-c^2$ $3.$ $a^2+b^2+c^2+d^2$ $4.$ ...
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0answers
335 views

Find the set of cyclotomic cosets of q modulo n

Calculating finite field and factoring $x^n - 1$ over $GF(q)$ first step is to calculate cyclotomic cosets. For example : For $n=9,q=2$ $C_1=\{1,2,4,8,7,5\} = C_4 = C_8 = C_7 = C_5$ $C_3=\{3,6\} ...
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0answers
377 views

Finding irreducible polynomials and factorization

Need some explanation and checking if my thinking on the solution is correct for the assignment given below: (In these problems you may use without proof which polynomials of degree 2 and 3 are ...
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0answers
39 views

Given a cubic $f(x)$ with specified negative real roots $-a,-b,-c$, what happens when we search for solutions to $f(x)=d$?

Noting Roots of a Certain type of Cubic Equation, what if we have the following simpler form for real $d$: $$(x+a)(x+b)(x+c)=d\tag{1}$$ (With $a,b,c\in \mathbb R^+$.) Clearly, depending on $d$, the ...
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0answers
71 views

Parametric Equation solving over integers

I have a question on my mind I am trying to solve. However I am stuck at a point. If you could help, I would be very pleased. $$\frac {x_2y_2z_2}{x_2y_2+y_2z_2+x_2z_2}=\frac ...
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0answers
51 views

Integer factorization using discrete logarithms

I'm reading up on RSA and attacks on it. At the end of one section of the notes, it asks (without giving an answer) whether or not integer factorization is easy given an oracle which computes discrete ...
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0answers
110 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 ...
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0answers
653 views

Factorization of cyclic polynomial

Factorize $a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)$ Since this is a cyclic polynomial therefore factors are also cyclic : $f(a) = a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)$ $f(b) = b(b^2-c^2)+b(c^2-b^2)+c(b^2-b^2) ...
1
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0answers
116 views

What is the condition for a polynomial to be factorizable in linear real factors?

I have a polynomial $p_a(x,y)= x^2F(a)+y^2G(a)-xH(a)-I(a)$ where $F(a)$, $G(a)$, $H(a)$ and $I(a)$ some real fuctions of $a$ are. Which conditions must satisfy $a$ so that I can factorize the ...
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0answers
82 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 ...
1
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0answers
138 views

A special factorization

Suppose that monic polynomial $f(x)\in\Bbb Z[x]$ such that for all $m\in\Bbb Z$, $m>1$, there's no integers $\langle r,r_1,\ldots,r_m\rangle$ such that $f(r)=f(r_1)\cdots f(r_m)$. Is there any ...
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0answers
104 views

Is there a name for a number whose factors' exponents are all prime?

For instance, 864, whose factorization is 2^5 x 3^3.
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272 views

Factoring multivariate polynomials

If I have a multivariate polynomial $P[X_1,\dots,X_n]\in \mathbb{R}[X_1,\dots, X_n]$, is there a polynomial time algorithm to factor the polynomial into irreducible polynomials $\in ...
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0answers
75 views

lower bounds for maximum computing times for integer factorisation

Supposing that n were known to have two prime factors, and that the computer had a database of all the primes $<\sqrt{n}$. Then, unless n is square, one factor would be $<\sqrt{n}$. If an ...
0
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0answers
24 views

Unable to get matched answer using factorization

I have question to solve by factorization. the question is $$(a+b)x^2 + (a+2b+c)x + (b+c) = 0$$ the answer should be $$x = -a, -b.$$ i have done using it \begin{align} (a+b)x^2 + (a+b+b+c)x + ...
0
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0answers
3 views

Factoriaztion of quasi homogeneous function

Let $f(x,y) \in C[x,y]$ be a quasi-homogeneous polynomial, with $f(t^{w_1}x,t^{w_1}y)=t^df(x,y)$ Supposedly, after an analytic change of variables, we can always write it as: $f(x,y) = ...
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0answers
25 views

Show a curve has no factor of degree 1 or 2

I have to show that $ h(x,y)=y^{2}(x^{2}+x+1)-x^{2} $ has no factors of degree 1 or 2. I know that h contains infinitely many points and is singular at the points (1,0,0), (0,1,0) and (0,0,1). I am ...
0
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0answers
30 views

Solving equation in maxima not placing variable on one side

I'm trying to solve an equation but the variable ($\varphi$ PHI) will not factor out to one side. Is there any other way to do this? I'm using maxima version 5.32.1 Here's the equation in latex as ...
0
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0answers
22 views

Integer Factorization

if $x \not \equiv \pm y$ (mod $n$) and $x^2 \equiv y^2$ (mod $n$), then $\gcd(x \pm y, n)$ are factors of $n$. Proof: $x^2 \equiv y^2$ (mod $n$) $\Rightarrow n$ is a factor of $(x-y)(x+y)$. Note ...
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0answers
19 views

Factoring big numbers into primes

I can't find a good tutorial anywhere on how to factor big numbers into primes, so I was wondering if someone could explain the process. I need to do this for my cryptology class.
0
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0answers
26 views

How to factor $(1-(i^2/n)(1/n)$ to isolate $i^2$ and form a sigma identity?

given sigma from $i=1$ to $n$ of $(1-(i^2/n)^2)(1/n))$ how would you factor this function to isolate $i^2$ and get $[n(n+1)(2n+1)]/6$ ? update... I got until the limit as n approaches infinity (1/n) ...
0
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0answers
60 views

Find factors of $0.08x^3 - 3.84x^2 + 42.66x - 137.7625$ using the Cubic Formula.

I have been going over this page as of late learning how to solve cubic formulas through depressing the equation, and solving for 'X'. Though, so far through numerous attempts, every single root I ...
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0answers
23 views

How to decompose N into A and B so that A and B are close to each other.

I would like to decompose $N$ into $A$ and $B$ so that $A$ and $B$ are close to each other. Even I would allow some small error. Se I would like to have: $$N = A*B + \epsilon_N$$ where $$\epsilon_N ...
0
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0answers
56 views

Positive integers of sum and products

Find all pairs of positive integers $m$ and $n$ where $m<n$ such that the sum of $m$ and $n$ added to the product of $m$ and $n$ is equal to $2014$ I just thought about this question and ...
0
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0answers
35 views

Quadratic Sieve Matrix Reduction

I have read through several other questions asked about this, but I want to be sure. Here are my current steps: Start with $\left[ \begin{array}{} 0 & 0 & 0 & 1\\ 1 & 1 & 1 ...
0
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0answers
20 views

Can I evaluate polynomials with prime numbers to find possible irreductible factors?

Let $p(x,y)$, $c(x,y)$ and $d(x,y)$ be two variable polynomials with integer coefficients which satisfy $p(x,y)=c(x,y)\cdot d(x,y)$. Given $m, n$ positive prime numbers and given $e(x,y)$ another ...
0
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0answers
16 views

Coppersmith method for factorisation

Is anyone familiar with the Coppersmith method? Does anyone know how is the $3\times3$ basis matrix obtained in this case?