Factorization Definition and 162 Threads

Hi, Can anyone confirm for me whether it has been proven that: if a number is congruent to 1 mod 4 and is expressed as the product of two factors, the difference between those factors will always be congruent to 0 mod 4; and that if the number is congruent to 3 mod 4 the difference between...

Hello everybody. I had been reading up on Unique Factorization again and I came across an interesting question. Can someone prove unique factorization for the set of polynomials in x, with integer coefficients? From what I understand, the analogous Euclidean algorithm works for such...

Homework Statement A is a 4 x 5 matrix equal to [1 4 -1 5 3 3 7 -2 9 6 -2 -3 6 -4 1 1 6 9 8 2] and b = [5 40 15 12] (b is 4 x 1) Find the LU factorization and use it to solve Ax = b Homework Equations The Attempt at a...

Figured it out.

Homework Statement Most invertible matrices can be written as a product A=LU of a lower triangular matrix L and an upper triangular matrix U, where in addition all diagonal entries of U are 1. a. Prove uniqueness, that is, prove that there is at most one way to write A as a product. b...

Homework Statement A positive integer a is called a square if a=n^2 for some n in Z. Show that the integer a>1 is a square iff every exponent in its prime factorization is even. Homework Equations The Attempt at a Solution Well, I know a=p1^a1p2^a2...pn^a^n is the definition of...

Homework Statement Prove unique factorization for hte set of polynomials in x with integer coefficients Homework Equations The Euclidean algorithm may be of some use The Attempt at a Solution Let's say that the polynomial is of the form anx^n + a(n-1)x^(n-1) ... a1x + a0 There...

Homework Statement Find the QR factorization for the 4x3 matrix M 1 1 0 1 0 2 1 0 1 1 1 1...

Homework Statement I'm trying to factorize a characteristic equation from my ODEs class, and I am having a problem when it comes to 4th, 5th or higher order differential equations. Say this one: r5-3r4+3r3-3r2+2r=0 Homework Equations The Attempt at a Solution Tried a lot of...

In my literature reviews I found a few things that I can't quite understand. Homework Statement I have the following equation: http://img717.yfrog.com/img717/6416/31771570.jpg I'm told that by using the eigenvalue factorization: http://img89.yfrog.com/img89/760/83769756.jpg , I can...

Homework Statement My book is awful and I need clarification on a few things regarding LU factorization: -If I am trying to express matrix A as a product of its upper triangular matrix (U) and the lower triangular matrix (L). I understand that I should find U first by Gauss-Jordon...

Homework Statement Assume that the matrix A is diagonalizable : A=PDP-1, where D is the diagonal matrix of eigenvalues. Show that this factorization is not always unique Homework Equations The Attempt at a Solution I have a couple of theories. The first being that since the...

Hi, This isn't exactly homework, but I guess it goes here because it's a specific problem? I'm working on some Fortran 90 code to simulate a quantum wire. Part of the calculation involves finding the determinant of the Hamiltonian. I'm currently doing this by calculating the Hamiltonian's LU...

Homework Statement You bring the drinks for your soccer team every sixth game. Every third game is a home game. How many times will you bring the drinks to a home game if you have a 20 game season? Homework Equations i did the problem in my head, but I need to show my work to get the...

Homework Statement In one part of a musical composition, the triangle player in an orchestra plays once every 12 beats. The tympani player plays once every 42 beats. How often do they play together? Homework Equations don't have any The Attempt at a Solution Insufficient...

Homework Statement Presidential elections are held every four years. Senators are elected every 6 years. If a senator was elected in the presidential election year of 2000, in what year would he or she campaign again during a presidential election year? Homework Equations dont know...

Homework Statement Margo has piano lessons every two weeks. Her brother Roberto has a soccer tournament every three weeks. Her sister Randa has an orthodontist appointment every four weeks. If they all have activities this Friday, how long will it be before all of their activities fall on...

given the Laplacian for a certain metric 'g' \Delta f = \partial _i \partial ^{i}f + (\partial ^{i}f) \partial _ i log |g|^{1/2} where a sum over 'i' dummy variable is assumed the idea is , could we factorize this Hamiltonian a second order differential operator into two first order...

Homework Statement in order to factorize 20 on the rings of Q( \sqrt 2) i must solve Homework Equations x^{2} -2y^{2}=10 The Attempt at a Solution i do not know how to solve it, i have tried by brute force with calculator but can not get any response , the given hint is that...

On average, at least how many factors must one try dividing a number N by to decompose it into primes?

Problem: Let Y1,Y2,...,Yn denote a random sample from the uniform distribution over the interval (0,theta). Show that Y(n)=max(Y1,Y2,...,Yn) is a sufficient statistic for theta by the factorization theorem. Solution: http://www.geocities.com/asdfasdf23135/stat10.JPG 1) While I...

Homework Statement Find the gcd of 22,471 and 3,266 and express in the form 22,471x + 3,266y Homework Equations The Attempt at a Solution I know how to get the gcd of easy numbers... using the prime factorization. But how do I do that with numbers of this scale?

Homework Statement 1. Homework Statement If n is a nonzero integer, prove that n cam be written uniquely in the form n=(2^k)m, where It is in the primes and unique factorization chapter so maybe that every integer n (except 0 and 1) can be written as a product of primes Homework...

Homework Statement If n is a nonzero integer, prove that n cam be written uniquely in the form n=(2^k)m, where k is greater than or equal to zero, and m is odd Homework Equations It is in the primes and unique factorization chapter so maybe that every integer n (except 0 and 1) can be...

I don't understand how to factor a polynomial over Z3 [x], Z7 [x], and Z11 [x] I need to factor the polynomail x3 - 23x2 - 97x + 291 PLEASE HELP!

Homework Statement One algorithm for finding the prime factorization of a number n is the following: Starting with d = 2, and continuing until n\geqd, try to divide n by d. If n/d, then record d as a (prime) factor and replace n by n/d; otherwise replace d by d + 1. a) When d is recorded...

Just curious

Trivial Question Involving Factorization (again) :) Homework Statement Factorize (E+M)(E+M) - as in E for energy and M for Mass Homework Equations ... n/a The Attempt at a Solution E^2+2ME+M^2 Thanks...

Homework Statement I have to write an algorithm in C++ to determine the (i)reductibility of a polynomial of degree "n" Homework Equations Berlekamp algorithm is preferred. The Attempt at a Solution I have Googled for almost an hour now and didn't find anything helpful. I...

Homework Statement Prove that \forall f:X\rightarrowY there \exists Z, h: X\rightarrowZ is injective and g: Z\rightarrowY is surjective, so that f=g*h. Homework Equations There is already a conclusion from the factorisation theorem of functions that: \forall f:X\rightarrowY there...

Alright I have to questions one is on how to measure the time it takes for my computer to solve a particular code I've tried the the "tic toc" and that seems to be dependent on the time frame that I typed in tic and toc. I need something that Is only dependent on the time taken to process and...

(I'm not sure what forum to put this on: number theory because of factoring, programming because I want to automate it, computers because I don't intend to actually program anything myself, or general because it combines these.) I'm looking for a program that I can use to partially factor...

I am not sure I fully understand the extension of the Unique Factorization Theorem (UFT) to Gaussian Integers (GI), by saying that the representation of a GI as a product of primes is unique except for the order of factors and the presence of units. Is there a similar problem when the UFT is...

I've been tasked with proving the existence of a full rank factorization for an arbitrary m x n matrix, namely: Let \textit{A} \in \textbf{R}^{m x n} with \textit{rank(A) = r} then there exist matrices \textit{B} \in \textbf{R}^{m x r} and \textit{C} \in \textbf{R}^{r x n} such that \textit{A =...

It occurred to me that the rationals Q have also a unique prime factorization, as long as you allow negative exponents on the factorization. If a/b is a rational, then both a and b have a unique (integer) prime factorization, and the fraction can be expressed uniquely as a product of primes...

I am posting an integer factorization algorithm that I have developed. I am hoping for feedback on any obvious flaws that I might have missed before writing a computer programme to test it out. Thanks in advance Visu

Factorize: x^3 − 2x^2 − 4x + 8 correct answer: (x^3 − 2x^2) − (4x − 8) x^2(x − 2) − 4(x − 2) (x^2 − 4)(x − 2) (x − 2)(x + 2)(x − 2) (x − 2) 2(x + 2) In the third line where the terms are grouped i don't understand why one of the (x - 2) is omitted? i.e shouldn't it be (x -2)^2 ?

I'm doing some basic factoring now to refresh my skills. On one problem, it is asking to factor: (y^2 + 8)^2 - 36 y^2 I was able to get to where I factor the trinomials: (y-4)(y-2)(y+4)(y+2) When it says to give the summary of factorization, it gives: (y^2 + 8)^2 - (6y)^2 I'm going...

For what values does \mathbb{Z}[\zeta] have unique factorization? I know Kummer shown that \zeta being a 23-rd root of unity fails to have unique factorization.

This is for a proof but I was generally more curious so it isn't in the homework section. If I were to make a set A which is defined as all the prime factors of an integer a there could be some numbers in A which are repeated, would these count as distinct members or not? The reason why I was...

Homework Statement Is there like a formula with a name easy to remember of which the following is a specific instance: x^200 -y^200 = (x-y)(x^199+x^198y+... + y^198*x + y^199) ? Homework Equations The Attempt at a Solution

This was taken from a math contest a few months ago. Homework Statement xx*yy=zz find z if: x=28 * 38 y=212 * 36 Homework Equations Theres undoubtably some trick, but I have yet to find it The Attempt at a Solution Dont even think about calculator I showed my math teacher, and...

Homework Statement Let A and B be invertible n×n matrices and b be an n×1 vector. Write a MATLAB function with inputs (A,B, b) to solve the equation x=B^−1*(2A^−1 + 1)b Make use of functions "LU Facotrization, Forward Substituion and Backwards Substitution" and DO NOT calculat any...

ok i know about the thing, i think its called gauss-jordan, where you do elimination on [A I] and you get [I A^-1] or [R E]or something like that. question is how can you get the E that puts A into reduced row echelon form with "row operations" only...supposedly some easy way ? on pg 134...

Hello everyone. I'm stuck on this problem and not sure how to apply the Unique factorization therem also called (Fundamental Theorem of Arithmetic). Heres the problem: http://suprfile.com/src/1/45ym0fx/lastscan.jpg The back of the book gives a pretty big hint which is the following...

This method utilizes discrete binary operations, in particular notice that. (I1) Div(x+y+z,a)=Div(x,a)+Div(y,a)+Div(z,a)+\{0,1,2\} This means that for one of the elements of the included set {0,1,2} the equation is true. This notation is quite useful for carrying out long calculations...

[Of course some of you will let me know if we need numerical examples or if something just does not make sense, or if it is very similar to what somebody else has already done, and I appreciate that.] Suppose we have a reducible polynomial over the natural numbers (zero include) such that...

for x^3 - 4x^2 -x = 0 , i have found one of the root which is 1 by dividing this equation by (x-1). from there onwards i can't do already to find the other two roots.somebody pls help thanx

Consider the following true statement. Given natural numbers n, m, s (1) \sum_{i=0}^n a^i = \sum_{i=0}^s a^{(m+1)i} \sum_{i=0}^m a^i iff (2) n+1=(m+1)(s+1) Now suppose that N is a natural number where (3) N = \sum_{i=0}^n a^i Now restrict a to the natural...

I haven't taken a math class since high school and I'm 23 now. I jumped right into Precalc 1 for the summer and the first chapter kicked my ass. I completely forgot how to factor and do LCD with algebraic equations and my professor just breezes by it like its nothing. Can anyone explain the...