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... Furthermore, all solutions of the Pell equation are of the form (pnL−1 , qnL−1 ), when L is even, and (p2nL−1 , q2nL−1 ) when L is odd, for n = 1, 2, . . . (cf. [10]). The geometrical locus containing all solutions of the Pell equation is the Pell hyperbola Hd = {(x, y) ∈ R2 : x2 − dy 2 = 1}. The fa ...
... Furthermore, all solutions of the Pell equation are of the form (pnL−1 , qnL−1 ), when L is even, and (p2nL−1 , q2nL−1 ) when L is odd, for n = 1, 2, . . . (cf. [10]). The geometrical locus containing all solutions of the Pell equation is the Pell hyperbola Hd = {(x, y) ∈ R2 : x2 − dy 2 = 1}. The fa ...
(pdf)
... previous case g −k ∈ hgi and hgi is closed under inverses. This covers all cases, so the desired equality has been established. It remains to show that |S| = ord(g). We first show this for the case when g has infinite order. Assume that for all n ∈ N, g n 6= e. If this is true, then we claim that fo ...
... previous case g −k ∈ hgi and hgi is closed under inverses. This covers all cases, so the desired equality has been established. It remains to show that |S| = ord(g). We first show this for the case when g has infinite order. Assume that for all n ∈ N, g n 6= e. If this is true, then we claim that fo ...
TGEA5 Chap 01
... One example is the square root of 2, written √2. It is the number that, when multiplied by itself, gives 2: √2 × √2 = 2. It can be shown that √2 cannot be written as a fraction with an integer numerator and an integer denominator. Therefore, it is not rational; it is an irrational number. It is inte ...
... One example is the square root of 2, written √2. It is the number that, when multiplied by itself, gives 2: √2 × √2 = 2. It can be shown that √2 cannot be written as a fraction with an integer numerator and an integer denominator. Therefore, it is not rational; it is an irrational number. It is inte ...
Учебно-методические материалы
... Proposition 1.1.11. Suppose that a and b are integers with b 6= 0. Then there exists unique integers q and r such that 0 r < jbj and a = bq + r. Proof. For simplicity, assume that both a and b are positive (we leave the general case to the reader). Let Q be the set of all nonnegative integers n such ...
... Proposition 1.1.11. Suppose that a and b are integers with b 6= 0. Then there exists unique integers q and r such that 0 r < jbj and a = bq + r. Proof. For simplicity, assume that both a and b are positive (we leave the general case to the reader). Let Q be the set of all nonnegative integers n such ...
Chapter4
... factorizations is not efficient because there is no efficient algorithm for finding the prime factorization of a positive integer. ...
... factorizations is not efficient because there is no efficient algorithm for finding the prime factorization of a positive integer. ...
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... Tk2.OK.2m 61 The Fn+2 compositions of (n + 1) using lfs and 2fs when put into the nested greatest integer function with 1 and 2 the exponents on a can be arranged so that the results are the integers 1, 2, . .., Fn + 2 i-n sequence. VtlOO^: We have illustrated Theorem 6 for n = 1, 2, . .., 5. Assume ...
... Tk2.OK.2m 61 The Fn+2 compositions of (n + 1) using lfs and 2fs when put into the nested greatest integer function with 1 and 2 the exponents on a can be arranged so that the results are the integers 1, 2, . .., Fn + 2 i-n sequence. VtlOO^: We have illustrated Theorem 6 for n = 1, 2, . .., 5. Assume ...
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.