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1: 27.9 Quadratic Characters
§27.9 Quadratic Characters
For an odd prime p , the Legendre symbol ( n | p ) is defined as follows. …If p does not divide n , then ( n | p ) has the value 1 when the quadratic congruence x 2 n ( mod p ) has a solution, and the value 1 when this congruence has no solution. … If p , q are distinct odd primes, then the quadratic reciprocity law states that … If an odd integer P has prime factorization P = r = 1 ν ( n ) p r a r , then the Jacobi symbol ( n | P ) is defined by ( n | P ) = r = 1 ν ( n ) ( n | p r ) a r , with ( n | 1 ) = 1 . …
2: 27.18 Methods of Computation: Primes
An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). … An alternative procedure is the binary quadratic sieve of Atkin and Bernstein (Crandall and Pomerance (2005, p. 170)). …
3: 24.14 Sums
§24.14(i) Quadratic Recurrence Relations
§24.14(ii) Higher-Order Recurrence Relations
4: 3.8 Nonlinear Equations
If p = 2 , then the convergence is quadratic; if p = 3 , then the convergence is cubic, and so on. … If ζ is a simple zero, then the iteration converges locally and quadratically. … It converges locally and quadratically for both and . … The method converges locally and quadratically, except when the wanted quadratic factor is a multiple factor of q ( z ) . … The quadratic nature of the convergence is evident. …
5: 15.17 Mathematical Applications
The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). …
6: 13.12 Products
For generalizations of this quadratic relation see Majima et al. (2000). …
7: 27.22 Software
  • 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 10 16 . Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard p 1 , 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).

  • 8: 16.6 Transformations of Variable
    Quadratic
    9: 27.19 Methods of Computation: Factorization
    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). …
    10: 18.7 Interrelations and Limit Relations
    §18.7(ii) Quadratic Transformations
    Equations (18.7.13)–(18.7.20) are special cases of (18.2.22)–(18.2.23). …