The factor-difference set of integers
... We wondered whether replacing the 8 factors in Na by more factors will yield examples of integers with more “small” differences. In order to succeed we will need to partition the 2n factors so that not only the sums of the weights will be equal but also higher order moments will be equal. Erdős rec ...
... We wondered whether replacing the 8 factors in Na by more factors will yield examples of integers with more “small” differences. In order to succeed we will need to partition the 2n factors so that not only the sums of the weights will be equal but also higher order moments will be equal. Erdős rec ...
Miles Reid's notes
... As we see many times later in the course, finding one root α of a polynomial f (x) in general only allows us to pull one factor x − α out of f , giving f (x) = (x − α)g(x). It certainly happens sometimes that g(x) is irreducible of degree n − 1 (see Example 3.23), and we have more work to do to fin ...
... As we see many times later in the course, finding one root α of a polynomial f (x) in general only allows us to pull one factor x − α out of f , giving f (x) = (x − α)g(x). It certainly happens sometimes that g(x) is irreducible of degree n − 1 (see Example 3.23), and we have more work to do to fin ...
Review/Outline Frobenius automorphisms Other roots of equations Counting irreducibles
... quintics are also primitive. In other words, α is a primitive root in F32 . And any (non-zero) element of F32 is of the form αt for some t in the range 1 ≤ t ≤ 31. Thus, we might try plugging α, α2 , α3 , etc into Q(x) to see whether we get 0. That is, replace x by x2 , x3 , x4 , etc and reduce modu ...
... quintics are also primitive. In other words, α is a primitive root in F32 . And any (non-zero) element of F32 is of the form αt for some t in the range 1 ≤ t ≤ 31. Thus, we might try plugging α, α2 , α3 , etc into Q(x) to see whether we get 0. That is, replace x by x2 , x3 , x4 , etc and reduce modu ...
Algebra 1B, Pre-Final Problems
... 63. Your quiz grades are 73, 75, 89, and 91. What is the lowest grade you can obtain on the last quiz and still achieve an average of at least 85? 64. Find the greatest possible pair of integers such that one integer is twice the other and their sum is less than 30. 65. The length of a rectangle is ...
... 63. Your quiz grades are 73, 75, 89, and 91. What is the lowest grade you can obtain on the last quiz and still achieve an average of at least 85? 64. Find the greatest possible pair of integers such that one integer is twice the other and their sum is less than 30. 65. The length of a rectangle is ...
Chapter 4
... PROPOSITION 4L. (EUCLID) There are infinitely many primes. Proof. Suppose on the contrary that p1 < . . . < pr are all the primes. Let n = p1 . . . pr + 1. Then n ∈ N and n > 1. It follows from the Fundamental theorem of arithmetic that pj | n for some j = 1, . . . , r, so that pj | (n − p1 . . . pr ...
... PROPOSITION 4L. (EUCLID) There are infinitely many primes. Proof. Suppose on the contrary that p1 < . . . < pr are all the primes. Let n = p1 . . . pr + 1. Then n ∈ N and n > 1. It follows from the Fundamental theorem of arithmetic that pj | n for some j = 1, . . . , r, so that pj | (n − p1 . . . pr ...
Section 4.1
... t b1 MOD a . The next example illustrates how Example 13 a special case illustrating how this problem is solved. Example 14: Consider a = 54321 and b = 9875 and consider the problem of solving bt 1 MOD a or 9875t 1 MOD 54321. In Example 13, we solved as + bt = gcd(a, b), and obtained t = -319 ...
... t b1 MOD a . The next example illustrates how Example 13 a special case illustrating how this problem is solved. Example 14: Consider a = 54321 and b = 9875 and consider the problem of solving bt 1 MOD a or 9875t 1 MOD 54321. In Example 13, we solved as + bt = gcd(a, b), and obtained t = -319 ...
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.