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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 … Both (27.9.1) and (27.9.2) are valid with $p$ replaced by $P$; the reciprocity law (27.9.3) holds if $p,q$ are replaced by any two relatively prime odd integers $P,Q$.
##### 2: 19.31 Probability Distributions
$R_{G}\left(x,y,z\right)$ and $R_{F}\left(x,y,z\right)$ occur as the expectation values, relative to a normal probability distribution in ${\mathbb{R}}^{2}$ or ${\mathbb{R}}^{3}$, of the square root or reciprocal square root of a quadratic form. …
##### 3: 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)). …
##### 5: 25.16 Mathematical Applications
which satisfies the reciprocity law
##### 6: 1.11 Zeros of Polynomials
The zeros of $z^{n}f(1/z)=a_{0}z^{n}+a_{1}z^{n-1}+\dots+a_{n}$ are reciprocals of the zeros of $f(z)$. …
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. …