CS173: Discrete Math
... prime is called composite • Fundamental theorem of arithmetic: Every positive integer greater than 1 can be written uniquely as a prime or as the product of two or more primes when the prime factors are written in order of non-decreasing size ...
... prime is called composite • Fundamental theorem of arithmetic: Every positive integer greater than 1 can be written uniquely as a prime or as the product of two or more primes when the prime factors are written in order of non-decreasing size ...
by Matthew Williamson
... the situation or , and let x, y = 0, 1, or 2. A single crossing would then be pictured as in figure 21. After an A-smoothing on the main k = 1 crossing, the tracked variables are n − 1, w, s − 1, rw − y, rs − 1, hw , and hs + js (figure 22), where js is a nonnegative integer. But, the situation in f ...
... the situation or , and let x, y = 0, 1, or 2. A single crossing would then be pictured as in figure 21. After an A-smoothing on the main k = 1 crossing, the tracked variables are n − 1, w, s − 1, rw − y, rs − 1, hw , and hs + js (figure 22), where js is a nonnegative integer. But, the situation in f ...
cs413encryptmathoverheads
... example of a commutative group than a non-commutative group. For instance, consider the positive and negative integers under addition. This satisfies all four of the requirements for a group and in addition the commutative property holds. In honor of the great Norwegian mathematician Niels Henrik Ab ...
... example of a commutative group than a non-commutative group. For instance, consider the positive and negative integers under addition. This satisfies all four of the requirements for a group and in addition the commutative property holds. In honor of the great Norwegian mathematician Niels Henrik Ab ...
Fraction
... We’ll use the GCF method first, and then the Prime Factorization method. GCF Method for Reducing Step 1: Find the GCF of the numerator and denominator Step 2: Factor the numerator and denominator using GCF Step 3: Cancel the GCF using the fact that a number over itself is always 1 Step 3: Rewrite th ...
... We’ll use the GCF method first, and then the Prime Factorization method. GCF Method for Reducing Step 1: Find the GCF of the numerator and denominator Step 2: Factor the numerator and denominator using GCF Step 3: Cancel the GCF using the fact that a number over itself is always 1 Step 3: Rewrite th ...
Prentice Hall Lesson ?
... • To simplify products, use the Distributive Property to multiply. (You may use FOIL if both expressions have two terms.) Combine like radicals and simplify. • Conjugates are the sum and difference of the same two terms. • The product of two conjugates is a difference of two squares. •To simplify a ...
... • To simplify products, use the Distributive Property to multiply. (You may use FOIL if both expressions have two terms.) Combine like radicals and simplify. • Conjugates are the sum and difference of the same two terms. • The product of two conjugates is a difference of two squares. •To simplify a ...
THE CONGRUENT NUMBER PROBLEM 1. Introduction
... Example 3.3. Taking n = 5 and (a, b, c) = (3/2, 20/3, 41/6), the correspondence in Theorem 3.1 yields (r, s, t) = (31/12, 41/12, 49/12): the rational squares (31/12)2 , (41/12)2 , (49/12)2 are an arithmetic progression with common difference 5. Example 3.4. Since Fermat showed 1 and 2 are not congru ...
... Example 3.3. Taking n = 5 and (a, b, c) = (3/2, 20/3, 41/6), the correspondence in Theorem 3.1 yields (r, s, t) = (31/12, 41/12, 49/12): the rational squares (31/12)2 , (41/12)2 , (49/12)2 are an arithmetic progression with common difference 5. Example 3.4. Since Fermat showed 1 and 2 are not congru ...
Factorization
In mathematics, factorization (also factorisation in some forms of British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained.The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viète's formulas relate the coefficients of a polynomial to its roots.The opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms.Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA.A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a surjective function with an injective function. This situation is generalized by factorization systems.