# quadratic

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## 1—10 of 39 matching pages

##### 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 ${x}^{2}\equiv n\phantom{\rule{0.949em}{0ex}}(modp)$ 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 $P={\prod}_{r=1}^{\nu \left(n\right)}{p}_{r}^{{a}_{r}}$, then the

*Jacobi symbol*$(n|P)$ is defined by $(n|P)={\prod}_{r=1}^{\nu \left(n\right)}{(n|{p}_{r})}^{{a}_{r}}$, with $(n|1)=1$. …

##### 2: 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).
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►An alternative procedure is the

*binary quadratic sieve*of Atkin and Bernstein (Crandall and Pomerance (2005, p. 170)). …##### 3: 24.14 Sums

##### 4: 3.8 Nonlinear Equations

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►If $p=2$, then the convergence is

*quadratic*; if $p=3$, then the convergence is*cubic*, and so on. … ►If $\zeta $ is a simple zero, then the iteration converges locally and quadratically. … ►It converges locally and quadratically for both $\mathbb{R}$ and $\u2102$. … ►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

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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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##### 6: 13.12 Products

##### 7: 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 ${10}^{16}$. 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

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###### Quadratic

…##### 9: 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). …