binary quadratic sieve
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1: 27.18 Methods of Computation: Primes
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►An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000).
►The Sieve of Eratosthenes (Crandall and Pomerance (2005, §3.2)) generates a list of all primes below a given bound.
An alternative procedure is the binary quadratic sieve of Atkin and Bernstein (Crandall and Pomerance (2005, p. 170)).
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2: 27.19 Methods of Computation: Factorization
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►These algorithms include the Continued Fraction Algorithm (cfrac), the Multiple Polynomial Quadratic Sieve (mpqs), the General
Number Field Sieve (gnfs), and the Special Number Field Sieve (snfs).
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3: 27.22 Software
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Mathematica. PrimeQ combines strong pseudoprime tests for the bases 2 and 3 and a Lucas pseudoprime test. No known composite numbers pass these three tests, and Bleichenbacher (1996) has shown that this combination of tests proves primality for integers below . Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard , and then cfrac after trial division. See §27.19. ecm is available also, and the Multiple Polynomial Quadratic sieve is expected in a future release.
For additional Mathematica routines for factorization and primality testing, including several different pseudoprime tests, see Bressoud and Wagon (2000).
4: 3.1 Arithmetics and Error Measures
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§3.1(i) Floating-Point Arithmetic
►Computer arithmetic is described for the binary based system with base 2; another system that has been used is the hexadecimal system with base 16. ►A nonzero normalized binary floating-point machine number is represented as … ►In the case of the normalized binary interchange formats, the representation of data for binary32 (previously single precision) (, , , ), binary64 (previously double precision) (, , , ) and binary128 (previously quad precision) (, , , ) are as in Figure 3.1.1. … ► …5: 27.9 Quadratic Characters
§27.9 Quadratic Characters
►For an odd prime , the Legendre symbol is defined as follows. …If does not divide , then has the value when the quadratic congruence has a solution, and the value when this congruence has no solution. … ►If are distinct odd primes, then the quadratic reciprocity law states that … ►If an odd integer has prime factorization , then the Jacobi symbol is defined by , with . …6: 24.14 Sums
7: 3.8 Nonlinear Equations
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►If , then the convergence is quadratic; if , then the convergence is cubic, and so on.
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►If is a simple zero, then the iteration converges locally and quadratically.
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►It converges locally and quadratically for both and .
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►The method converges locally and quadratically, except when the wanted quadratic factor is a multiple factor of .
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►The quadratic nature of the convergence is evident.
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8: 15.17 Mathematical Applications
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►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations.
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►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)).
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9: 13.12 Products
10: 16.6 Transformations of Variable
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