Questions tagged [factoring]

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

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Coefficients of some polynomials factorization

Let $n,b\in\mathbb{N}$ and $P_{n_b}$ is polynomial whose coefficients are digits of $n$ in base $b$. Let $Q_{1}(x) \cdot Q_{2}(x) \cdots Q_{m}(x)$ - polynomial factorization (over integers) of $P_{n_b}...
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Factorizing $AMA^T+N=WW^T$ efficiently.

This question is related to this question with a slightly more general setting. A good speedup on this could improve performances of a decision process in multi-objective optimization that I designed. ...
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What is the relation between these two descriptions of General Number Field Sieve?

I am reading about General Number Field Sieve (GNFS) algorithm and its implementations, and the literature seems to have two different versions of the algorithm, but I can't find how these two are ...
Sil's user avatar
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Find all $x \in \mathbb{R}$ such that $1+\sqrt{1-\frac{1}{x}}+\sqrt{x-\frac{1}{x}}=x^2$.

This problem was posed by a friend of mine. He's got it from a social media post. We don't know how to solve it, but there are similar types of questions in this website, whose techniques I've tried ...
asd3weqsda zfewrewgsfd's user avatar
2 votes
1 answer
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How to factorize $2^{3^{k}}+1$

My question: How do we get $2^{3^{k}}+1=(2+1)\left(2^{2}-2+1\right)\left(2^{2 \cdot 3}-2^{3}+1\right) \ldots\left(2^{2 \cdot 3^{k-1}}-2^{3^{k-1}}+1\right)$$(k\ge 0)$ I know $2^{3^{k}}+1=(2+1)(2^{3^{k}-...
Penguin's user avatar
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Factoring $x^2 + 4x -3$ step by step.

I have the following equation that I need to factor: $x^2 + 4x -3$. I cannot use the factoring by grouping method as there are no integers that add up to $4$ and give $-3$ when multiplied. What method ...
Paul's user avatar
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Factoring $x^2 - 2 + \frac1{x^2}$ by steps

$$x^2 - 2 + \frac1{x^2}$$ factors out to $$\left(x-\frac1x\right)^2$$ I know this because the answer was given to me in a video. If the answer wasn't given to me, I would have been stuck. I'm not ...
VennDiagram's user avatar
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1 answer
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Dscriminant for a poynomial of degree higher than 3

I am dealing with a polynomial of degree 5, which I want to prove is positive on a certain interval. I found a statement on Wikipedia that exactly enable me to do this. There is no reference backing ...
Gateau au fromage's user avatar
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How to find factors of a number that add to a certain sum?

For example, when solving this equation for $x$ $x^{2}+8x-9=0$ I would write $x^2-x+9x-9=0$ Then write that expression as the product of two linear expressions: $(x+9)(x-1)=0$ Then I solve the two ...
Will Fitchet's user avatar
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Which type of "numbers" will require order 3 factors for its irreducible polynomial factorizations?

The complex numbers are famous for having the property of the fundamental theorem of algebra. All polynomials have at least one root. Together with being factor-able, this means we can factor every ...
mathreadler's user avatar
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Product of two non-negative polynomials has coefficients 0, 1, or 2 [closed]

A 0-1-2 polynomial is a univariate polynomial where all coefficients are either $0$, $1$, or $2$. Is it true that if two real polynomials $P(x), Q(x)$, have their product equal to a 0-1-2 polynomial (...
mick's user avatar
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Induction argument to show that $x-\lambda$ and $x-\bar{\lambda}$ appear the same number of times in the factorization of $p\in P(\mathbb{R})$?

Consider a polynomial $p\in\mathcal{P}(\mathbb{R})$ with a complex root $\lambda$. Since the coefficients of $p$ are real, we know that complex roots occur in conjugate pairs. We can write $p(x)$ as $$...
xoux's user avatar
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How to see $2n^2 + 7n +6 = (n+2)(2n+3)$?

In a proof I was reading of the sum of the first $n$ squares formula, the fact is used that $2n^2 + 7n +6 = (n+2)(2n+3)$. I was struggling with the exact logic used in deducing this relation- I see we ...
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Exact factorization of $a^m+b^m$ and $a^m-b^m$

Let $a,b$ be two indeterminates. I read here that $a^m + b^m$ is divisible by $(a+b)$ if $m$ is odd and $a^m - b^m$ is divisible by $(a+b)$ if $m$ is even. I am interested in an exact expression $f(a,...
motionart's user avatar
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Factoring bivariate polynomials

Consider the following bivariate polynomial $$P (x,y) = \sum_{m=0}^{2}\sum_{n=0}^{2} a_{m,n} x^{m} y^{n}$$ where the coefficients $a_{m,n}$ can be complex numbers. I was wondering if there are any ...
XZhao's user avatar
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How can this sinh curve be massaged?

Sorry I'm new here so I don't know how to use the math symbols yet but I would really appreciate some help with something I'm stuck on. How can this equation: $$ 𝑦(𝑥)=𝑥 \sinh(\sinh^{-1}(𝑦_0)−\rho \...
Mr_Ryder's user avatar
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Factorization of $f(x)=x$ in $\Bbb{Z}/n\Bbb{Z}[x]$ where $n$ is the product of $k$ distinct primes (considering $\Bbb{Z}/30\Bbb{Z}[x]$ as example)

Intro This is the final part of the problem 9.4.20 (e), (d) from Dummit and Foote's Abstract Algebra. They formulate it as "Determine all the factorizations of $f(x)=x$ in $\Bbb{Z}/n\Bbb{Z}[x]$ ...
Stanarth's user avatar
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Factoring a year to get a date

Let's factor 2024: $2024=8\times 11\times 23$. So we can make a date: 8/11/23, or August 11, 2023. Or we could also make 11/8/23, November 8, 2023 (if you live in Europe, they will be switched). ...
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Factoring integers by finding nontrivial divisors

Most integer factoring algorithms work by finding a nontrivial divisor of their input. In order to fully factor a given input, algorithms recursively apply the procedure for finding a nontrivial ...
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Factorising a Cubic Polynomial

I was studying cubic factorisation and had just begun to get the hang of it. Cubic equations which are of the form $$ax^3+bx^2+cx+d:\frac{a}{b}=\frac{c}{d}$$ are naturally straightforward to solve, ...
Schrödinger's Cat's user avatar
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What are some useful factoring tips and tricks for math contests and competitions? [closed]

What are some useful advanced factoring tips and tricks for someone interested in participating in an olympiad style math contest? I am already familiar with taking the common factor, grouping like ...
MushroomTea's user avatar
2 votes
2 answers
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Finding positive integer $n>10$ that maximizes $\frac{\sigma_0(n)}{2^{\log n}}$

Among all the positive integer, which one integer, $n$, can make the number below the largest? $$f(n)=\frac{\sigma_0(n)}{2^t}$$where $t=\log_{10}n$ and $\sigma_0$ is the divisor function. For example,...
A Math guy's user avatar
3 votes
2 answers
144 views

Are there any efficient methods to manually factor a quadratic in two variables?

For a quadratic in two variables, $ax^2+bxy+cy^2+dx+ey+f$ may be factored into the form $(p_1x+q_1y+r_1)(p_2x+q_2y+r_2)$ given that $\det\begin{bmatrix}a & b/2 & d/2\\ b/2 & c & e/2\\d/...
Catherine's user avatar
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Factoring $2 + \frac{4}{2Z + 1} - \frac{3}{Z} + \frac{Z}{2Z^2 - Z}$

How to factor below expression please? $$2 + \frac{4}{2Z + 1} - \frac{3}{Z} + \frac{Z}{2Z^2 - Z}$$ What I got is $$\frac{8Z^3 - 2Z^2 - 5Z + 3}{Z(4Z^2 - 1)}$$ but I was told the answer doesn't look ...
ddss12's user avatar
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If sum of the series $5+7+11+19+35+67+131....$ upto 10 terms is $x$, then number of factors of $x$ between $1$ and $10$ is

If sum of the series $5+7+11+19+35+67+131+....$ upto 10 terms is $x$, then number of factors of $x$ between $1$ and $10$ is My solution: Since first difference of given series in Geometric ...
mathophile's user avatar
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Finding a Prime Using Sin Function

To find if n is prime, take the square root of n, called m. Let O be the set of primes less than or equal to m. For every element o in O, let y = sin⁡(πx/o), for all real numbers x from zero to n. So ...
fwmbowers's user avatar
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Why does using $f(x)=x^2$ not work in Pollard's rho algorithm?

In Pollard's rho algorithm for integer factorization, we use pseudorandom sequences of the form $x_{i+1}=f(x_i)$ and look at them$\mod{n}$ until we get a cycle. Utilizing the birthday paradox, we can ...
shp's user avatar
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Has anyone studied factoring as a CO-product?

In factorization, like integer factorization, you start with an integer and end up with a kind-of list of pairs of other elements, namely the factors. I want to explore the "Co-ness" of this....
Ben Sprott's user avatar
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1 answer
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How to factorize the polynomial $x^4+3x^2+x$ [closed]

$$x^4+3x^2+x=0$$ I only can think of $$x(x^3+3x+1)$$ I don't know what else to do if substitution fails and also RRT. Is this solvable? or we have to use imaginary solutions?
samsamradas's user avatar
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2 answers
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Is there any faster way to factorize this polynomial $p(x) = x^3-8x^2+19x-12$?

Let $p(x) = x^3-8x^2+19x-12$. By RRT it follows that $(x-1)$ is a factor. After that doing long division I get $$(x-1)(x^2-7x+12)$$ $$(x-1)(x-3)(x-4).$$ But is there any faster way to do it?
samsamradas's user avatar
1 vote
1 answer
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factoring the equation for union of lines algebraically

For example, we have a quadratic polynomial $F(x,y)$ and the graph of $F(x,y)=0$ are the union of two intersecting lines. Now, how do we show that $F(x,y)=k(a_1x+b_1y+c_1)(a_2x+b_2y+c_2)$ ? My ...
ZhenRanZR's user avatar
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A question about the factors of reducible multilinear polynomial.

Suppose that $P\in\mathbb{R}[x_1,\ldots,x_n]$ is a multilinear polynomial and suppose that $$ P=Q_1Q_2\cdots Q_n $$ where each $Q_i\in\mathbb{R}[x_1,\ldots,x_n]$ is non-constant. Comment: We say that ...
boaz's user avatar
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2 votes
2 answers
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What kind of a formula is this and does it belong to combinatorics?

$$ x^n-y^n $$ Hi I have a limit to calculate with this formula and I don't know where it belongs,I only know the binomial combination.Is this formula related to combinatorics? $$\lim_{x \to 1} \frac {...
marceldrak's user avatar
9 votes
1 answer
1k views

How is GNFS the best factoring algorithm when its time complexity exceeds brute-force?

While researching another problem, the Sieve of Eratosthenes got me wondering why it couldn’t be used to factor. After playing around, I stumbled upon an algorithm that seems like it should find ...
Russ Johnson's user avatar
2 votes
0 answers
278 views

Can factoring $90$ help factor $91$?

There are few posts asking if factoring $N-1$ can help factor $N$. In those posts the focus was on the factors of $N-1$ and $N$ which can never be the same. The conclusion therefore was that ...
user25406's user avatar
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Quadratic Formula False?!?!?!?!

The world is over. Quadratic formula finds intercepts correctly but not the right graph. What am I missing? If $0=2x^2+7x+5$, then it can be solved with the quadratic formula to give us $x=-5/2$ or $-...
Rufus Gordon-Heywood's user avatar
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Can I factor a binomial coefficient out from an equation?

By the distributive property, if I'm given the equation XY + XZ I can factor this to X(Y +Z). I'm wondering if I could apply this same property to a binomial coefficient, e.g., I'm given nCr(n, X) * Y ...
Ryan Panek's user avatar
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1 answer
64 views

Find all irreducible factors of a polynomial over $\mathbb{Z}$

I am trying to solve this question for my abstract algebra class. Let $f = X^6 - 2X^5 + 3X^4 - 2X^3 + 3X^2 - 2X + 2 \in \mathbb{Z}[X]$. Either prove that $f$ is irreducible over $\mathbb{Z}$ or find ...
Zek's user avatar
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Factorize a sum of matrix products into a product of linear factors

Let $A$, $B$, $C$, $D$, $E$ be rectangular real matrices of appropriate sizes, and define a psd matrix $S$ by $$ S = CB AA'B'C' + DD' + EE'. $$ Question 1. How to write $S$ as a product $$ S := ...
dohmatob's user avatar
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Finding an algorithm to factor $n$ given cube root $\mod n$

Let $p,q$ be unknown primes and $n=pq$. Also let: $p\equiv 4 \mod 9$ $q\equiv 4 \mod 9$ Imagine I have an "oracle" that takes cube roots $\mod n$. Find a probabilistic algorithm to factor $n$...
Cotton Headed Ninnymuggins's user avatar
1 vote
1 answer
132 views

Factorisation of the sum of $n$ squares using hypercomplex numbers

After finding out that you could factor $a^2 + b^2$ as $(a+bi)(a-bi)$ using complex numbers I wondered if there were any useful factoring tricks using the quaternions or octonions and after some ...
hefe's user avatar
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-1 votes
1 answer
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Is there a way to factor: $x^2 - 6xy + y^2$?

Is there a way to factor $x^2 - 6xy + y^2$? I know the result, if there is one, would be irrational. Also, for cases like this where the result would include irrational terms like $\sqrt{2}$, is there ...
Ryan Pierce Williams's user avatar
0 votes
1 answer
158 views

How do I factorise an infinite binomial expansion with negative index

So, I was wondering how could I factorise an infinite binomial expansion, say, $$(1-2x)^{-1} = 1 + 2 x + 4 x^2 + 8 x^3 + 16 x^4 + \dots$$ I know that there exist methods to factorise infinite ...
Poke_Programmer's user avatar
0 votes
1 answer
51 views

Intuition while factorizing an equation

I have the following expression to factorize: $x^4 - 15x^2 + 36$ One way to solve it is: $x^4 - 15x^2 + 36$ $x^4 - 12x^2 - 3x^2 + 36$ $x^2 (x^2 - 12) -3 (x^2 - 12)$ $(x^2 - 12) (x^2 -3)$ $(x-\sqrt{12})...
Imtiaz's user avatar
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1 answer
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Find all possible values of the perimeter of a rectangle with sides $x, y \in \mathbb{Z}^+$, where its area is given by $A=3x+3y+\sqrt{9x^2+9y^2}$

Find all possible values of the perimeter of a rectangle with sides positive integers $x$ and $y$ where its area is given by $A=3x+3y+\sqrt{9x^2+9y^2}$ In other words if $xy=3x+3y+\sqrt{9x^2+9y^2}$, ...
sinichgaja's user avatar
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1 answer
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In what book can I find the proof of the identity for factoring a difference of similar powers? (Referring to the identity $x^n-a^n = (x-a) (\ldots)$)

I have found this identity $x^{n} - a^{n}= (x - a)(x^{n-1} + ax^{n-2} + \dots + a^{n-1})$ while reading Calculus by James Stewart, where it is needed to solve an exercise about partial fractions. ...
JPPM's user avatar
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1 answer
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My answer in long division and synthetic division dont match

my problem was $2x^4 + x^3 -19x^2 + 18x + 5$ divided by $2x-5$ my answer in long division was : $x^3 + 3x^2 - 2x + 4$ and a remainder of $25$ my answer in synthetic division : $2x^3 + 11x^2 + 36x + ...
Louisf Nobleza's user avatar
0 votes
1 answer
68 views

How to factorize the above expression?

How to factorize $x^4+x+7$? I am trying this question by adding and subtracting $x^2$. So, the expression is becoming $(x^4+x^2+x+7)-x^2$. Now I am trying to factorize $x^2+x+7$ first. So, therefore $...
Shivam Kumar's user avatar
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0 answers
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Are all Superior Highly Composite Numbers multiples of all smaller Superior Highly Composite Numbers?

Looking at the Wikipedia entry on the topic: Superior Highly Composite Numbers It is true of all numbers listed there. Does this commonality continue forever? Put another way: Are all SHCNs factors of ...
Georgina Davenport's user avatar
3 votes
1 answer
300 views

Center Binomial can "Self-Factor" RSA semiprimes. [closed]

Given two odd primes $P$ and $Q$ s.t. $P < Q$, $PQ = N$, $P \neq 3$ and $Q \neq 5$ at same time... the center binomial $f(k)=\frac{(2k)!}{(k!)^{2}}$ can be used to factor $N$. This will not be ...
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