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

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2
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3answers
145 views

irreducibility of polynomials with integer coefficients

Consider the polynomial $$p(x)=x^9+18x^8+132x^7+501x^6+1011x^5+933x^4+269x^3+906x^2+2529x+1733$$ Is there a way to prove irreducubility of $p(x)$ in $\mathbb{Q}[x]$ different from asking to PARI/GP?
0
votes
1answer
55 views

Factorising and limits

How do I factorize this expression? $$(2^n-3^n+n4^n)^{\frac{1}{n}}$$ so far I have: $$n4^n\left(\frac{1}{n} \left(\frac{1}{2}\right)^n-\frac{1}{n}\left(\frac{3}{4}\right)^n +1\right)^{\frac{1}{n}}$$ ...
4
votes
3answers
143 views

Factoring $x^8-x^4+1$ over $GF(7)$

Could anyone suggest any good way to do it? (The only way I can think of is by looking for roots (There are none), checking a factorization into the product of a 6 and a 2 polynomial (Many unknowns ...
7
votes
1answer
462 views

factorise, $x^3-13x^2+32x+20$

factorise, $x^3-13x^2+32x+20$ Let, $f(x)=x^3-13x^2+32x+20$ $f(x)=x(x^2-13x+30)+2x+20$ $f(x)=x(x-3)(x-10)+2x+20$ $f(-1)\lt 0$, $f(0)\gt 0$, which shows there is a root between $x=-1$ and $x=0$ ...
6
votes
2answers
224 views

Simple factoring in proof by induction

How would this: $$\frac{((n+1)+1)(2(n+1) + 1)(2(n+1) + 3)}{3}$$ Factor to this: $$(2(n+1)+1)^2$$ This is a part of an induction proof, which I would post an image if my reputation was higher... ...
9
votes
1answer
168 views

Factoring a couple $5$th degree polynomials

I'm reading an old (1895) textbook on algebra (doing a bit of review), and practicing factoring polynomials. The author started with polynomials where all terms share a common factor, like $4a^2 + 4a ...
1
vote
3answers
116 views

Trouble with factorising a polynomial

I'm supposed to show that: $$y=\frac{5(x-1)(x+2)}{(x-2)(x+3)} = P + \frac{Q}{(x-2)} + \frac{R}{(x+3)}$$ and the required answers are: $$ P=5, Q=4, R=-4 $$ I tried to solve this with partial ...
1
vote
2answers
307 views

Factorization of three variables

Prove that : $(a+b+c)^3-(b+c)^3-(c+a)^3-(a+b)^3+a^3+b^3+c^3=6abc$ Since $(a+b+c)^3=a^3+b^3+c^3+3(a+b)(b+c)(c+a) $ Therefore the equation becomes : $2(a^3+b^3+c^3)+3(a+b)(b+c)(c+a) - [(c+a)^3 ...
1
vote
4answers
115 views

Factoring a large integer where the factors are prime

Suppose I know the value of $n$ (where $n = pq$), and I also know $k = p-q$. How can I efficiently factor $n$? Note that I don't know $p$ or $q$. EDIT: Thank you for your answers. I understand that ...
1
vote
2answers
70 views

Factoring polynomial with 4 terms

I cant figure this out. $$\frac{x-5}{x^3-3x^2+7x-5}$$ I tried by grouping and got $$x^2(x-3)+1(7x-5)$$ for the denominator. I need to use partial fractions on this so I cant use that yet.
4
votes
1answer
168 views

Does the difficulty of discrete logarithm depend on the difficulty of integer factorization?

The security of many (most? all?) public-key cryptography systems are based on the difficulty of the discrete logarithm or integer factorization. Are these two problems related at all? With the ...
0
votes
1answer
53 views

What is $q(x)$ and $r(x)$ when $(x^2-6x+9)q(x) + r(x) = x^3 -27$?

I just failed this question on a test, so I would please like to get some feedback on where my thinking was wrong. I need help with determining $q(x)$ and $r(x)$ when: $$(x^2-6x+9)q(x) + r(x) = x^3 ...
3
votes
1answer
90 views

Factorization of integers - why does it suffice to consider squarefree instances?

I sat a lecture where a proposition is proven that states the following: If computation of $(k!)_{k\in\mathbb{N}}$ is "easy", then integer numbers can be factored in non uniform polynomial time. ...
1
vote
0answers
676 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) ...
2
votes
1answer
142 views

Help with particular solution to solving $z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$.

I asked a question about how to solve: $$z^4−2z^3+9z^2−14z+14 = 0$$ When all you know is that there is a root with the real part of 1. I was given great answers and you can find the question ...
7
votes
3answers
613 views

How do I completely solve the equation $z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$ where there is a root with the real part of $1$.

I would please like some help with solving the following equation: $$z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$$ All I know about the equation is that there is a root with the real part of $1$. My approach ...
1
vote
3answers
234 views

finding residue with $\oint_C \dfrac{3z^3 + 2}{(z-1)(z^2 + 9)} dz$

I am doing the integral $\oint_C \dfrac{3z^3 + 2}{(z-1)(z^2 + 9)} dz$, and I am trying to find the residue at the pole $3i$;I am unsure how to do this. Could I factor $z^2 + 9$ further?
0
votes
1answer
150 views

Integer Factoring Algorithm Speeds

Given $N=pq$, would $\frac{p-1}{2}$ steps be fast compared with extant factoring methods?
2
votes
1answer
1k views

Using Parseval' s theorem to evaluate a sum..

On the function $f(x)$ = $x^3$ on $(-1,1)$ find Fourier coefficients for this function and then use Parseval's Theorem to evaluate: $$\sum_{n=1}^{\infty}\frac{1}{n^6}$$ Current work: I have used ...
0
votes
1answer
99 views

Gaussian elimination with partial pivoting doubts

I have the following doubts about Gauss algorithm with partial pivoting: Say that I sum to the second row the first row multiplied by $k$. In the $L$ matrix, should I sum to the second row the first ...
0
votes
1answer
36 views

Distinct-degree factorization

I'm trying to understand distinct-degree factorization from Wikipedia. I'm trying the algorithm on paper with $q=9$ and $f(x) = (x+4)(x+5) = x^2+2 \in F_{q}$. We start with $i=1$. I calculate $g = ...
4
votes
5answers
139 views

Polynomial factoring $1-3x+4x^3$

I want to factorize (or factor ? can both verbs be used ?) $1-3x+4x^3$. I notice that $\frac{1}{2}$ and $-1$ are roots of the polynomial. My questions are : 1) how do you notice that $\frac{1}{2}$ ...
0
votes
1answer
23 views

What is the pair $(n,k)$ called where $n$ is an integer and $k$ is the ordered factorization index?

I’m developing a number class (as in Object-Oriented Programming) and am wondering what to call it. At its core, it represents an integer, but in a way in which not all integers are unique. What it ...
2
votes
3answers
224 views

Solve for $ x$, $-\frac{1}{2}x^2 + 2x + 5 = 0$

I'm having trouble solving this equation for $x$: $$-\frac{1}{2}x^2 + 2x + 5 = 0$$ What's the steps to take to solve it? Thanks.
5
votes
3answers
6k views

How many positive integers are factors of a given number?

I've been trying to find / generate a formula for the following problem: Given a number, how many positive integers are factors of this number. In practice, you could simply build a table as such ...
3
votes
1answer
106 views

Limit question factoring [duplicate]

Possible Duplicate: How do I find the delta analytically for $f(x)$ with a degree other than $1$ There is a question, prove that: $$\displaystyle \lim_{x\to3} x^{2} = 9$$ in linear ...
1
vote
1answer
99 views

Can the product of irreducible polynomials have non-constant factors other than those polynomials?

Can the product of irreducible polynomials over the reals, $P_1, P_2,...,P_n$, have non-constant polynomial factors other than those polynomials or products of them (eg. $P_1P_3$)? It seems that the ...
1
vote
1answer
85 views

legal value for prime factorization

My understanding of mathematics is poor, although I'm trying to improve it, so I hope you will forgive me for asking a rather basic question. I've written a computer code that prints the prime ...
0
votes
1answer
47 views

solving an equation by factoring

Need help solving an equation by factoring. PROBLEM: $3v^2-10v-12 = -28v + 36$ This is my solution, but it seems a little too much for school: $3v^2+18v-48 = 0$ $3(v^2+18v-48) = 0$ ...
3
votes
1answer
197 views

Matrix Processing in the Quadratic Sieve

I am working through the example in implementation of the quadratic sieve, and I have got stuck at the very last part: the matrix processing. In the example we must find the vector $S$ by left null ...
6
votes
1answer
1k views

Simply Explain the General Number Field Sieve

As a beginner to the world of integer factorization, my idea of factoring an integer is to generate a large list of prime numbers below this number and to repeatedly try to divide the integer by these ...
1
vote
3answers
94 views

Find the value of $\frac{1}{d_1}+\frac{1}{d_2}+\dots+\frac{1}{d_k}$.

Let $d_1,d_2,\dots,d_k$ be all the factors of a positive integer $n$ including $1$ and $n$. Suppose $d_1+d_2+\dots+d_k=72$. Then the value of $$\frac{1}{d_1}+\frac{1}{d_2}+\dots+\frac{1}{d_k}$$ is ...
1
vote
1answer
70 views

probability of a number not having factors below n?

I want to know the probability that a number is not divisible by any prime smaller than or equal to n. ie, I find a big number, I trial division it up to 300000 and don't find any factors. I want to ...
2
votes
5answers
4k views

Factoring Polynomials with four terms and two variables

I've been working on this for hours and cannot figure it out. When I search, I find factorization techniques that I already know but don't seem to be able to apply here, or that are for polynomials ...
1
vote
2answers
223 views

How to break apart this sum?

I have a summation I need to break apart but I can't figure it out http://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape10/PQDD_0027/MQ50799.pdf $p.15$, right after line $(3.8)$ Going from the ...
5
votes
3answers
106 views

Given $N$, find $ab = N$ with $a$ and $b$ as close as possible

Given a number $N$ I would like to factor it as $N=ab$ where $a$ and $b$ are as close as possible; say when $|b-a|$ is minimal. For certain $N$ this is trivial: when $N$ is prime, a product of two ...
2
votes
2answers
84 views

Multiplying over Subtraction

I'd like to take maths seriously but I'm not that great at it, so I decided to learn at home. I know this is pretty basic, but as I said, I'm pretty bad at maths, haha. The worksheet gives me this ...
1
vote
0answers
121 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 ...
0
votes
2answers
74 views

Transform a positive integer to find its next greatest factor

Suppose I am trying to find factors of a particular positive integer num. Suppose I also have a function findGreatestFactor(num) ...
1
vote
1answer
190 views

Factoring a polynomial with big integer coefficients and some known factors.

I have the following polynomial that I want to factor $$ \begin{align*} p(x)= &- 236364091 x^{13}- 28363690920 x^{12}- 1487737229594 x^{11}\\ &- 44880832661940 x^{10} - 860924276925225 x^9- ...
18
votes
4answers
788 views

Find all roots of $\,(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$

The question is to find all complex roots of $$(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$$ and it is meant to be solved by hand. Is there any quick way to solve this using some trick that I'm not ...
16
votes
1answer
436 views

Irreducibility of $x^{n}+x+1$

Motivated by this problem, and KCd's comment on my answer, I am left with the following question: Question: Suppose that $n\not \equiv 2\pmod{3}$. Is $$x^n+x+1$$ irreducible over $\mathbb{Q}$? ...
0
votes
1answer
245 views

Zeroes in a 3x3 Matrix Determinant

My professor found the cubic roots of a 3x3 matrix by doing the following. I don't understand how step 2 came about and why he applied the same for step 4 on row 1 instead of row 2. Step 1: ...
4
votes
1answer
161 views

How to implement birthday paradox continuation of elliptic curve factorization algorithm

I have already implemented Lenstra's algorithm for factoring integers using elliptic curves; it is shown below, or you can run it at http://ideone.com/QEDmMY. Beware that my code is optimized for ...
1
vote
1answer
49 views

factorization of numbers with euclidian approach

Anyone know about this topic?. Factorization of numbers with euclidian approach I searching in the internet, but i couldnt find any source of this topic?. Some one can help me about this topic?. I ...
11
votes
7answers
519 views

Can someone show me why this factorization is true?

$$x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + \dots + xy^{n-2} + y^{n-1})$$ Can someone perhaps even use long division to show me how this factorization works? I honestly don't see anyway to "memorize ...
0
votes
3answers
85 views

How to factorize a cubic equation?

How should I factor this polynomial: $x^3 - x^2 - 4x - 6$
1
vote
1answer
67 views

Polynomial factoring issue

I am dealing with an issue for which I do not find answer on the Internet. When I factorize a polynomial, I can get this structure: $$ (x-a)(x-b)(x-c)^2 $$ But sometimes I have seen others like: $$ ...
1
vote
1answer
253 views

Complexity of Pollard's p-1 method

I'm working on the complexity of various integer factorization algorithms and am kind of stuck on the complexity of Pollard's p-1 method. (I'm using Prime Numbers - A Computational Perspective by ...
2
votes
0answers
50 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 ...