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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 $x^{2}\equiv n\pmod{p}$ 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 $P=\prod_{r=1}^{\nu\left(n\right)}p^{a_{r}}_{r}$, 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
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)). …
##### 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 $\zeta$ is a simple zero, then the iteration converges locally and quadratically. … It converges locally and quadratically for both $\mathbb{R}$ and $\mathbb{C}$. … 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
Complete cases of Legendre’s integrals and symmetric integrals can be computed with quadratic convergence by the AGM method (including Bartky transformations), using the equations in §19.8(i) and §19.22(ii), respectively. … As $n\to\infty$, $c_{n}$, $a_{n}$, and $t_{n}$ converge quadratically to limits $0$, $M$, and $T$, respectively; hence … Computation of Legendre’s integrals of all three kinds by quadratic transformation is described by Cazenave (1969, pp. 128–159, 208–230). Quadratic transformations can be applied to compute Bulirsch’s integrals (§19.2(iii)). …