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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).