Editing Quadratic residue (section)

==Notations==

Gauss<ref>Gauss, DA, art. 131</ref> used {{math|R}} and {{math|N}} to denote residuosity and non-residuosity, respectively;
:for example, {{math|2 R 7}} and {{math|5 N 7}}, or {{math|1 R 8}} and {{math|3 N 8}}.

Although this notation is compact and convenient for some purposes,<ref>e.g. Hardy and Wright use it</ref><ref>Gauss, DA, art 230 ff.</ref> a more useful notation is the [[Legendre symbol]], also called the [[Dirichlet character#Examples|quadratic character]], which is defined for all integers {{mvar|a}} and positive odd [[prime number]]s {{mvar|p}} as
:<math>
\left(\frac{a}{p}\right) = \begin{cases}\;\;\,0&\text{ if }p \text { divides } a\\+1&\text{ if } a \operatorname{R} p \text{ and }p \text { does not divide } a\\-1&\text{ if }a \operatorname{N} p \text{ and }p \text{ does not divide } a\end{cases}</math>

There are two reasons why numbers ≡ 0 (mod {{mvar|p}}) are treated specially. As we have seen,  it makes many formulas and theorems easier to state. The other (related) reason is that the quadratic character is a [[homomorphism]] from the [[multiplicative group of integers modulo n|multiplicative group of nonzero congruence classes modulo {{mvar|p}}]] to the [[complex numbers]] under multiplication. Setting <math>(\tfrac{np}{p}) = 0</math> allows its [[Domain of a function|domain]] to be extended to the multiplicative [[semigroup]] of all the integers.<ref>This extension of the domain is necessary for defining ''L'' functions.</ref>

One advantage of this notation over Gauss's is that the Legendre symbol is a function that can be used in formulas.<ref>See [[Legendre symbol#Properties of the Legendre symbol]] for examples</ref>  It can also easily be generalized to [[Cubic reciprocity|cubic]], quartic and higher power residues.<ref>Lemmermeyer, pp 111&ndash;end</ref>

There is a generalization of the Legendre symbol for composite values of {{mvar|p}}, the [[Jacobi symbol]], but its properties are not as simple: if {{mvar|m}} is composite and the Jacobi symbol <math>(\tfrac{a}{m}) = -1,</math> then {{math|''a'' N ''m''}}, and if {{math|''a'' R ''m''}} then <math>(\tfrac{a}{m}) = 1,</math> but if <math>(\tfrac{a}{m}) = 1</math> we do not know whether {{math|''a'' R ''m''}} or {{math|''a'' N ''m''}}. For example: <math>(\tfrac{2}{15}) = 1</math> and <math>(\tfrac{4}{15}) = 1</math>, but {{math|1=2 N 15}} and {{math|1=4 R 15}}. If {{mvar|m}} is prime, the Jacobi and Legendre symbols agree.