Primetime
... tiles does it take to make each rectangle? Explain your reasoning. b. What is the length of each of Kyong’s rectangles? Explain your reasoning. c. Without changing the number of tiles used to make either rectangle, Kyong rearranges the tiles of each rectangle into different rectangles. What is a pos ...
... tiles does it take to make each rectangle? Explain your reasoning. b. What is the length of each of Kyong’s rectangles? Explain your reasoning. c. Without changing the number of tiles used to make either rectangle, Kyong rearranges the tiles of each rectangle into different rectangles. What is a pos ...
Rational points on Shimura curves and Galois representations Carlos de Vera Piquero
... of curves over both number fields and their completions. To some extent, the goal of this work can be therefore framed into the more general problem of describing the set of rational points of an algebraic variety over a number field or a local field1. Suppose for simplicity that X is a smooth proje ...
... of curves over both number fields and their completions. To some extent, the goal of this work can be therefore framed into the more general problem of describing the set of rational points of an algebraic variety over a number field or a local field1. Suppose for simplicity that X is a smooth proje ...
IDEAL FACTORIZATION 1. Introduction We will prove here the
... has no prime factorization then let n > 1 be minimal without a prime factorization. Of course n is not prime, so n = ab with a, b > 1. Then a, b < n, so a and b are products of primes. Hence n = ab is a product of primes, which is a contradiction. Uniqueness of the prime factorization requires more ...
... has no prime factorization then let n > 1 be minimal without a prime factorization. Of course n is not prime, so n = ab with a, b > 1. Then a, b < n, so a and b are products of primes. Hence n = ab is a product of primes, which is a contradiction. Uniqueness of the prime factorization requires more ...
Reducing Fractions
... Whole numbers are used for counting; that is, describing the number of objects in a group. However, the result of a measurement need not be a whole number, and in fact, rarely is. The number of pages in this book is by definition a whole number, but the weight of the book in pounds is probably not a ...
... Whole numbers are used for counting; that is, describing the number of objects in a group. However, the result of a measurement need not be a whole number, and in fact, rarely is. The number of pages in this book is by definition a whole number, but the weight of the book in pounds is probably not a ...
IDEAL FACTORIZATION 1. Introduction
... has no prime factorization then let n > 1 be minimal without a prime factorization. Of course n is not prime, so n = ab with a, b > 1. Then a, b < n, so a and b are products of primes. Hence n = ab is a product of primes, which is a contradiction. Uniqueness of the prime factorization requires more ...
... has no prime factorization then let n > 1 be minimal without a prime factorization. Of course n is not prime, so n = ab with a, b > 1. Then a, b < n, so a and b are products of primes. Hence n = ab is a product of primes, which is a contradiction. Uniqueness of the prime factorization requires more ...
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.