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| <div class="thumb tright"> | | <br><br> |
| <div class="thumbinner" style="width:220px;"> | |
| {| class="wikitable" style="width:220px;"
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| ! {{nobreak|p \ a}} !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10
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| |-
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| ! 3
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| |bgcolor="#FFFF00"| 0
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| |bgcolor="#FFFF00"| 1 || -1 || || || || || || || ||
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| |-
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| ! 5
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| |bgcolor="#FFFF00"| 0
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| |bgcolor="#FFFF00"| 1 || -1 || -1
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| |bgcolor="#FFFF00"| 1 || || || || || ||
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| |-
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| ! 7
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| |bgcolor="#FFFF00"| 0
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| |bgcolor="#FFFF00"| 1
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| |bgcolor="#FFFF00"| 1 || -1
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| |bgcolor="#FFFF00"| 1 || -1 || -1 || || || ||
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| |-
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| ! 11
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| |bgcolor="#FFFF00"| 0
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| |bgcolor="#FFFF00"| 1 || -1
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| |bgcolor="#FFFF00"| 1
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| |bgcolor="#FFFF00"| 1
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| |bgcolor="#FFFF00"| 1 || -1 || -1 || -1
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| |bgcolor="#FFFF00"| 1 || -1
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| |}
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| <div class="thumbcaption">
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| Legendre symbol (a/p) for various ''a'' (along top) and ''p'' (along left side). Only 0 ≤ ''a'' < ''p'' are shown, since due to the first property below any other ''a'' can be reduced modulo ''p''. [[Quadratic residue]]s are highlighted in yellow, and correspond precisely to the values 0 and 1.
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| </div>
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| </div>
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| </div>
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| In [[number theory]], the '''Legendre symbol''' is a [[multiplicative function]] with values 1, −1, 0 that is a quadratic character modulo a [[prime number]] ''p'': its value on a (nonzero) [[quadratic residue]] mod ''p'' is 1 and on a non-quadratic residue (''non-residue'') is −1. Its value on zero is 0.
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|
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|
| The Legendre symbol was introduced by [[Adrien-Marie Legendre]] in 1798<ref>A. M. Legendre ''Essai sur la theorie des nombres'' Paris 1798, p 186</ref> in the course of his attempts at proving the [[law of quadratic reciprocity]]. Generalizations of the symbol include the [[Jacobi symbol]] and [[Dirichlet character]]s of higher order. The notational convenience of the Legendre symbol inspired introduction of several other "symbols" used in [[algebraic number theory]], such as the [[Hilbert symbol]] and the [[Artin symbol]].
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| == Definition ==
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| Let ''p'' be an odd [[prime number]]. An integer ''a'' is a [[quadratic residue]] modulo ''p'' if it is [[modular arithmetic|congruent]] to a [[square number|perfect square]] modulo ''p'' and is a quadratic nonresidue modulo ''p'' otherwise. The '''Legendre symbol''' is a function of ''a'' and ''p'' defined as follows:
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| :<math>
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| \left(\frac{a}{p}\right) =
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| \begin{cases}
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| \;\;\,1 \text{ if } a \text{ is a quadratic residue modulo}\ p
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| \text{ and } a \not\equiv 0\pmod{p} \\
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| -1 \text{ if } a \text{ is a quadratic non-residue modulo}\ p\\
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| \;\;\,0 \text{ if } a \equiv 0 \pmod{p}.
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| \end{cases}
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| </math> | |
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| Legendre's original definition was by means of an explicit formula:
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| :<math> \left(\frac{a}{p}\right) \equiv a^{(p-1)/2}\ \pmod{ p}\;\;\text{ and } \left(\frac{a}{p}\right) \in \{-1,0,1\}.</math>
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| By [[Euler's criterion]], which had been discovered earlier and was known to Legendre, these two definitions are equivalent.<ref>Hardy & Wright, Thm. 83.</ref> Thus Legendre's contribution lay in introducing a convenient ''notation'' that recorded quadratic residuosity of ''a'' mod ''p''. For the sake of comparison, [[Carl Friedrich Gauss|Gauss]] used the notation <math>a\mathrm{R}p</math>, <math>a\mathrm{N}p</math> according to whether ''a'' is a residue or a non-residue modulo ''p''.
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| For typographical convenience, the Legendre symbol is sometimes written as (''a''|''p'') or (''a''/''p''). The sequence (''a''|''p'') for ''a'' equal to 0,1,2,... is [[periodic sequence|periodic]] with period ''p'' and is sometimes called the '''Legendre sequence''', with {0,1,−1} values occasionally replaced by {1,0,1} or {0,1,0}.<ref name=KimSong01>Jeong-Heon Kim and Hong-Yeop Song, "Trace Representation of Legendre Sequences," ''Designs, Codes, and Cryptography'' '''24''', p. 343–348 (2001).</ref>
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| == Properties of the Legendre symbol ==
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| There are a number of useful properties of the Legendre symbol which, together with the law of [[quadratic reciprocity]], can be used to compute it efficiently.
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| * The Legendre symbol is periodic in its first (or top) argument: if ''a'' ≡ ''b'' (mod ''p''), then
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| ::<math>
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| \left(\frac{a}{p}\right) = \left(\frac{b}{p}\right).
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| </math>
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| * The Legendre symbol is a [[completely multiplicative function]] of its top argument:
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| ::<math>
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| \left(\frac{ab}{p}\right) = \left(\frac{a}{p}\right)\left(\frac{b}{p}\right).
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| </math>
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| : In particular, the product of two numbers that are both quadratic residues or quadratic non-residues modulo ''p'' is a residue, whereas the product of a residue with a non-residue is a non-residue. A special case is the Legendre symbol of a square:
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| ::<math>
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| \left(\frac{x^2}{p}\right) =
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| \begin{cases} 1&\mbox{if }p\nmid x\\0&\mbox{if }p\mid x.
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| \end{cases}
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| </math>
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| * When viewed as a function of ''a'', the Legendre symbol <math>\left(\frac{a}{p}\right)</math> is the unique quadratic (or order 2) [[Dirichlet character]] modulo ''p''.
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| * The first supplement to the law of quadratic reciprocity:
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| ::<math>
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| \left(\frac{-1}{p}\right)
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| = (-1)^\tfrac{p-1}{2}
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| =\begin{cases}
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| \;\;\,1\mbox{ if }p \equiv 1\pmod{4} \\
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| -1\mbox{ if }p \equiv 3\pmod{4}. \end{cases}</math>
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| * The second supplement to the law of quadratic reciprocity:
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| ::<math>
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| \left(\frac{2}{p}\right)
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| = (-1)^\tfrac{p^2-1}{8}
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| =\begin{cases}
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| \;\;\,1\mbox{ if }p \equiv 1\mbox{ or }7 \pmod{8} \\
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| -1\mbox{ if }p \equiv 3\mbox{ or }5 \pmod{8}. \end{cases}</math>
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| * Special formulas for the Legendre symbol <math>\left(\frac{a}{p}\right)</math> for small values of ''a'':
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| :* For an odd prime ''p'' ≠ 3,
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| ::<math>
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| \left(\frac{3}{p}\right)
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| = (-1)^{\big\lfloor \frac{p+1}{6}\big\rfloor}
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| =\begin{cases}
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| \;\;\,1\mbox{ if }p \equiv 1\mbox{ or }11 \pmod{12} \\
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| -1\mbox{ if }p \equiv 5\mbox{ or }7 \pmod{12}. \end{cases}</math>
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| :* For an odd prime ''p'' ≠ 5,
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| ::<math>
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| \left(\frac{5}{p}\right)
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| =(-1)^{\big\lfloor \frac{p+2}{5}\big \rfloor}
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| =\begin{cases}
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| \;\;\,1\mbox{ if }p \equiv 1\mbox{ or }4 \pmod5 \\
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| -1\mbox{ if }p \equiv 2\mbox{ or }3 \pmod5. \end{cases}</math>
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| * The [[Fibonacci numbers]] 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... are defined by the recurrence {{nowrap|F<sub>1</sub> {{=}} F<sub>2</sub> {{=}} 1,}} {{nowrap|F<sub>n+1</sub> {{=}} F<sub>n</sub> + F<sub>n−1</sub>.}} If ''p'' is a prime number then
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| :<math>
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| F_{p-\left(\frac{p}{5}\right)} \equiv 0 \pmod p,\;\;\;
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| F_{p} \equiv \left(\frac{p}{5}\right) \pmod p.
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| </math>
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| <blockquote>
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| For example,
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| :<math>
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| \begin{align}
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| &(\tfrac{2}{5}) &= &-1, &F_3 &= 2, \;\;\;\;&F_2 &=1,\\
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| &(\tfrac{3}{5}) &= &-1, &F_4 &= 3,&F_3&=2,\\
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| &(\tfrac{5}{5}) &= &\;\;\;\;0, &F_5 &= 5,\\
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| &(\tfrac{7}{5}) &= &-1, &F_8 &= 21,&F_7&=13,\\
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| &(\tfrac{11}{5}) &= &\;\;\,1, &F_{10} &= 55, &F_{11}&=89.
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| \end{align}
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| </math>
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| This result comes from the theory of [[Lucas sequence]]s, which are used in [[primality testing]].<ref>Ribenboim, p. 64; Lemmermeyer, ex 2.25-2.28, pp. 73–74.</ref> See [[Wall–Sun–Sun prime]].
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| </blockquote>
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| == Legendre symbol and quadratic reciprocity ==
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| Let ''p'' and ''q'' be odd primes. Using the Legendre symbol, the [[quadratic reciprocity]] law can be stated concisely:
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| : <math>
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| \left(\frac{q}{p}\right)
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| = \left(\frac{p}{q}\right)(-1)^{\tfrac{p-1}{2}\tfrac{q-1}{2}}.
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| </math>
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| Many [[proofs of quadratic reciprocity]] are based on Legendre's formula
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| :<math>
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| \left(\frac{a}{p}\right) \equiv a^{\tfrac{p-1}{2}}\pmod p.
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| </math>
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| In addition, several alternative expressions for the Legendre symbol were devised in order to produce various proofs of the quadratic reciprocity law.
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| * Gauss introduced the [[quadratic Gauss sum]] and used the formula
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| ::<math>
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| \sum_{k=0}^{p-1}\zeta^{ak^2}=
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| \left(\frac{a}{p}\right)\sum_{k=0}^{p-1}\zeta^{k^2},
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| \quad \zeta = e^{2\pi i/p}
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| </math>
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| :in his fourth<ref>Gauss, "Summierung gewisser Reihen von besonderer Art" (1811), reprinted in ''Untersuchungen ...'' pp. 463–495</ref> and sixth<ref>Gauss, "Neue Beweise und Erweiterungen des Fundamentalsatzes in der Lehre von den quadratischen Resten" (1818) reprinted in ''Untersuchungen ...'' pp. 501–505</ref> proofs of quadratic reciprocity.
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| * [[Leopold Kronecker|Kronecker's]] proof<ref>Lemmermeyer, ex. p. 31, 1.34</ref> first establishes that
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| ::<math>
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| \left(\frac{p}{q}\right)
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| =\sgn\left(\prod_{i=1}^{\frac{q-1}{2}}\prod_{k=1}^{\frac{p-1}{2}}\left(\frac{k}{p}-\frac{i}{q}\right)\right).</math>
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| : Reversing the roles of ''p'' and ''q'', he obtains the relation between <math>\left(\frac{p}{q}\right)</math> and <math>\left(\frac{q}{p}\right).</math>
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| * One of [[Gotthold Eisenstein|Eisenstein]]'s proofs<ref>Lemmermeyer, pp. 236 ff.</ref> begins by showing that
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| ::<math>
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| \left(\frac{q}{p}\right)
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| =\prod_{n=1}^{\frac{p-1}{2}} \frac{\sin\left(\frac{2\pi qn}{p}\right)}{\sin\left(\frac{2\pi n}{p}\right)}.
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| </math>
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| : Using certain [[elliptic function]]s instead of the [[sine function]], Eisenstein was able to prove [[cubic reciprocity|cubic]] and [[quartic reciprocity|quartic]] reciprocity as well.
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| == Related functions ==
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| * The [[Jacobi symbol]] <math>\left(\tfrac{a}{n}\right)</math> is a generalization of the Legendre symbol that allows for a composite second (bottom) argument ''n'', although ''n'' must still be odd and positive. This generalization provides an efficient way to compute all Legendre symbols without performing factorization along the way.
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| * A further extension is the [[Kronecker symbol]], in which the bottom argument may be any integer.
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| * The [[power residue symbol]] <math>\left(\tfrac{a}{p}\right)_n</math> generalizes the Legendre symbol to higher power ''n''. The Legendre symbol represents the [[power residue symbol]] for ''n'' = 2.
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| == Computational example ==
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| The above properties, including the law of quadratic reciprocity, can be used to evaluate any Legendre symbol. For example:
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| :<math>\left ( \frac{12345}{331}\right )</math>
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| :<math>=\left ( \frac{3}{331}\right ) \left ( \frac{5}{331}\right ) \left ( \frac{823}{331}\right )</math>
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| :<math>=\left ( \frac{3}{331}\right ) \left ( \frac{5}{331}\right ) \left ( \frac{161}{331}\right )</math>
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| :<math>=\left ( \frac{3}{331}\right ) \left ( \frac{5}{331}\right ) \left ( \frac{7}{331}\right ) \left ( \frac{23}{331}\right )</math>
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| :<math>= (-1) \left ( \frac{331}{3}\right ) \left ( \frac{331}{5}\right ) (-1) \left ( \frac{331}{7}\right ) (-1) \left ( \frac{331}{23}\right )</math>
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| :<math>=-\left ( \frac{1}{3}\right ) \left ( \frac{1}{5}\right ) \left ( \frac{2}{7}\right ) \left ( \frac{9}{23}\right )</math>
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| :<math>=-\left ( \frac{1}{3}\right ) \left ( \frac{1}{5}\right ) \left ( \frac{2}{7}\right ) \left ( \frac{3^2}{23}\right )</math>
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| :<math>= - \left (1\right ) \left (1\right ) \left (1\right ) \left (1\right ) = -1.</math>
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| Or using a more efficient computation:
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| :<math>\left ( \frac{12345}{331}\right )=\left ( \frac{98}{331}\right )=\left ( \frac{2 \cdot 7^2}{331}\right )=\left ( \frac{2}{331}\right )=(-1)^\tfrac{331^2-1}{8}=-1.</math>
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| The article [[Jacobi symbol#Calculations_using_the_Legendre_symbol|Jacobi symbol]] has more examples of Legendre symbol manipulation.
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| | |
| *{{citation
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| | last1 = Gauss | first1 = Carl Friedrich
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| | last2 = Maser | first2 = H. (translator into German)
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| | title = Untersuchungen über höhere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition)
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| | publisher = Chelsea
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| | location = New York
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| | year = 1965
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| | isbn = 0-8284-0191-8}}
| |
| | |
| *{{citation
| |
| | last1 = Gauss | first1 = Carl Friedrich
| |
| | last2 = Clarke | first2 = Arthur A. (translator into English)
| |
| | title = Disquisitiones Arithmeticae (Second, corrected edition)
| |
| | publisher = [[Springer Science+Business Media|Springer]]
| |
| | location = New York
| |
| | year = 1986
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| | isbn = 0-387-96254-9}}
| |
| | |
| *{{citation
| |
| | last1 = Bach | first1 = Eric
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| | last2 = Shallit | first2 = Jeffrey
| |
| | title = Algorithmic Number Theory (Vol I: Efficient Algorithms)
| |
| | publisher = [[The MIT Press]]
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| | location = Cambridge
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| | year = 1996
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| | isbn = 0-262-02405-5}}
| |
| | |
| *{{citation
| |
| | last1 = Hardy | first1 = G. H.
| |
| | authorlink1=G. H. Hardy
| |
| | last2 = Wright | first2 = E. M.
| |
| | title = An Introduction to the Theory of Numbers (Fifth edition)
| |
| | publisher = [[Oxford University Press]]
| |
| | location = Oxford
| |
| | year = 1980
| |
| | isbn = 978-0-19-853171-5}}
| |
| | |
| *{{citation
| |
| | last1 = Ireland | first1 = Kenneth
| |
| | last2 = Rosen | first2 = Michael
| |
| | title = A Classical Introduction to Modern Number Theory (Second edition)
| |
| | publisher = [[Springer Science+Business Media|Springer]]
| |
| | location = New York
| |
| | year = 1990
| |
| | isbn = 0-387-97329-X}}
| |
| | |
| *{{citation
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| | last1 = Lemmermeyer | first1 = Franz
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| | title = Reciprocity Laws: from Euler to Eisenstein
| |
| | publisher = [[Springer Science+Business Media|Springer]]
| |
| | location = Berlin
| |
| | year = 2000
| |
| | isbn = 3-540-66957-4}}
| |
| | |
| *{{citation
| |
| | last1 = Ribenboim | first1 = Paulo
| |
| | title = The New Book of Prime Number Records
| |
| | publisher = [[Springer Science+Business Media|Springer]]
| |
| | location = New York
| |
| | year = 1996
| |
| | isbn = 0-387-94457-5}}
| |
| | |
| == External links ==
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| *[http://www.math.fau.edu/richman/jacobi.htm Jacobi symbol calculator]
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| [[Category:Modular arithmetic]]
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| [[Category:Quadratic residue]]
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