Legendre symbol

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Legendre symbol (a/p)
for various a (along top) and p (along left side).
a
p
0 1 2 3 4 5 6 7 8 9 10
3 0 1 −1
5 0 1 −1 −1 1
7 0 1 1 −1 1 −1 −1
11 0 1 −1 1 1 1 −1 −1 −1 1 −1

Only 0 ≤ a < p are shown, since due to the first property below any other a can be reduced modulo p. Quadratic residues are highlighted in yellow, and correspond precisely to the values 0 and 1.

In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd 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.

The Legendre symbol was introduced by Adrien-Marie Legendre in 1798[1] in the course of his attempts at proving the law of quadratic reciprocity. Generalizations of the symbol include the Jacobi symbol and Dirichlet characters 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.

Definition[edit]

Let p be an odd prime number. An integer a is a quadratic residue modulo p if it is congruent to a perfect square modulo p and is a quadratic nonresidue modulo p otherwise. The Legendre symbol is a function of a and p defined as

Legendre's original definition was by means of the explicit formula

By Euler's criterion, which had been discovered earlier and was known to Legendre, these two definitions are equivalent.[2] Thus Legendre's contribution lay in introducing a convenient notation that recorded quadratic residuosity of a mod p. For the sake of comparison, Gauss used the notation aRp, aNp according to whether a is a residue or a non-residue modulo p.

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 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}.[3]

Table of values[edit]

The following is a table of values of Legendre symbol (k/n) with n ≤ 127, k ≤ 30, n odd prime.

k
n
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
3 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0
5 1 −1 −1 1 0 1 −1 −1 1 0 1 −1 −1 1 0 1 −1 −1 1 0 1 −1 −1 1 0 1 −1 −1 1 0
7 1 1 −1 1 −1 −1 0 1 1 −1 1 −1 −1 0 1 1 −1 1 −1 −1 0 1 1 −1 1 −1 −1 0 1 1
11 1 −1 1 1 1 −1 −1 −1 1 −1 0 1 −1 1 1 1 −1 −1 −1 1 −1 0 1 −1 1 1 1 −1 −1 −1
13 1 −1 1 1 −1 −1 −1 −1 1 1 −1 1 0 1 −1 1 1 −1 −1 −1 −1 1 1 −1 1 0 1 −1 1 1
17 1 1 −1 1 −1 −1 −1 1 1 −1 −1 −1 1 −1 1 1 0 1 1 −1 1 −1 −1 −1 1 1 −1 −1 −1 1
19 1 −1 −1 1 1 1 1 −1 1 −1 1 −1 −1 −1 −1 1 1 −1 0 1 −1 −1 1 1 1 1 −1 1 −1 1
23 1 1 1 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1 1 −1 1 −1 −1 −1 −1 0 1 1 1 1 −1 1 −1
29 1 −1 −1 1 1 1 1 −1 1 −1 −1 −1 1 −1 −1 1 −1 −1 −1 1 −1 1 1 1 1 −1 −1 1 0 1
31 1 1 −1 1 1 −1 1 1 1 1 −1 −1 −1 1 −1 1 −1 1 1 1 −1 −1 −1 −1 1 −1 −1 1 −1 −1
37 1 −1 1 1 −1 −1 1 −1 1 1 1 1 −1 −1 −1 1 −1 −1 −1 −1 1 −1 −1 −1 1 1 1 1 −1 1
41 1 1 −1 1 1 −1 −1 1 1 1 −1 −1 −1 −1 −1 1 −1 1 −1 1 1 −1 1 −1 1 −1 −1 −1 −1 −1
43 1 −1 −1 1 −1 1 −1 −1 1 1 1 −1 1 1 1 1 1 −1 −1 −1 1 −1 1 1 1 −1 −1 −1 −1 −1
47 1 1 1 1 −1 1 1 1 1 −1 −1 1 −1 1 −1 1 1 1 −1 −1 1 −1 −1 1 1 −1 1 1 −1 −1
53 1 −1 −1 1 −1 1 1 −1 1 1 1 −1 1 −1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 −1 −1 1 1 −1
59 1 −1 1 1 1 −1 1 −1 1 −1 −1 1 −1 −1 1 1 1 −1 1 1 1 1 −1 −1 1 1 1 1 1 −1
61 1 −1 1 1 1 −1 −1 −1 1 −1 −1 1 1 1 1 1 −1 −1 1 1 −1 1 −1 −1 1 −1 1 −1 −1 −1
67 1 −1 −1 1 −1 1 −1 −1 1 1 −1 −1 −1 1 1 1 1 −1 1 −1 1 1 1 1 1 1 −1 −1 1 −1
71 1 1 1 1 1 1 −1 1 1 1 −1 1 −1 −1 1 1 −1 1 1 1 −1 −1 −1 1 1 −1 1 −1 1 1
73 1 1 1 1 −1 1 −1 1 1 −1 −1 1 −1 −1 −1 1 −1 1 1 −1 −1 −1 1 1 1 −1 1 −1 −1 −1
79 1 1 −1 1 1 −1 −1 1 1 1 1 −1 1 −1 −1 1 −1 1 1 1 1 1 1 −1 1 1 −1 −1 −1 −1
83 1 −1 1 1 −1 −1 1 −1 1 1 1 1 −1 −1 −1 1 1 −1 −1 −1 1 −1 1 −1 1 1 1 1 1 1
89 1 1 −1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 1 1 1 −1 −1 1 −1 −1 −1 −1 −1
97 1 1 1 1 −1 1 −1 1 1 −1 1 1 −1 −1 −1 1 −1 1 −1 −1 −1 1 −1 1 1 −1 1 −1 −1 −1
101 1 −1 −1 1 1 1 −1 −1 1 −1 −1 −1 1 1 −1 1 1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 1
103 1 1 −1 1 −1 −1 1 1 1 −1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 1 −1 1 1 −1 1 1 1
107 1 −1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 −1 1 −1 −1 1 −1 −1 −1 1 −1 1 −1 1 −1 1 1
109 1 −1 1 1 1 −1 1 −1 1 −1 −1 1 −1 −1 1 1 −1 −1 −1 1 1 1 −1 −1 1 1 1 1 1 −1
113 1 1 −1 1 −1 −1 1 1 1 −1 1 −1 1 1 1 1 −1 1 −1 −1 −1 1 −1 −1 1 1 −1 1 −1 1
127 1 1 −1 1 −1 −1 −1 1 1 −1 1 −1 1 −1 1 1 1 1 1 −1 1 1 −1 −1 1 1 −1 −1 −1 1

Properties of the Legendre symbol[edit]

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.

  • The Legendre symbol is periodic in its first (or top) argument: if ab (mod p), then
  • 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:
  • When viewed as a function of a, the Legendre symbol (a/p) is the unique quadratic (or order 2) Dirichlet character modulo p.
  • The first supplement to the law of quadratic reciprocity:
  • The second supplement to the law of quadratic reciprocity:
  • Special formulas for the Legendre symbol (a/p) for small values of a:
    • For an odd prime p ≠ 3,
    • For an odd prime p ≠ 5,
  • The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... are defined by the recurrence F1 = F2 = 1, Fn+1 = Fn + Fn−1. If p is a prime number then
For example,

Legendre symbol and quadratic reciprocity[edit]

Let p and q be distinct odd primes. Using the Legendre symbol, the quadratic reciprocity law can be stated concisely:

Many proofs of quadratic reciprocity are based on Legendre's formula

In addition, several alternative expressions for the Legendre symbol were devised in order to produce various proofs of the quadratic reciprocity law.

in his fourth[5] and sixth[6] proofs of quadratic reciprocity.
Reversing the roles of p and q, he obtains the relation between (p/q) and (q/p).
Using certain elliptic functions instead of the sine function, Eisenstein was able to prove cubic and quartic reciprocity as well.

Related functions[edit]

  • The Jacobi symbol (a/n) 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.
  • A further extension is the Kronecker symbol, in which the bottom argument may be any integer.
  • The power residue symbol (a/n)n generalizes the Legendre symbol to higher power n. The Legendre symbol represents the power residue symbol for n = 2.

Computational example[edit]

The above properties, including the law of quadratic reciprocity, can be used to evaluate any Legendre symbol. For example:

Or using a more efficient computation:

The article Jacobi symbol has more examples of Legendre symbol manipulation.

Notes[edit]

  1. ^ Legendre, A. M. (1798). Essai sur la théorie des nombres. Paris. p. 186. 
  2. ^ Hardy & Wright, Thm. 83.
  3. ^ Kim, Jeong-Heon; Song, Hong-Yeop (2001). "Trace Representation of Legendre Sequences". Designs, Codes, and Cryptography. 24: 343–348. 
  4. ^ Ribenboim, p. 64; Lemmermeyer, ex. 2.25–2.28, pp. 73–74.
  5. ^ Gauss, "Summierung gewisser Reihen von besonderer Art" (1811), reprinted in Untersuchungen ... pp. 463–495
  6. ^ Gauss, "Neue Beweise und Erweiterungen des Fundamentalsatzes in der Lehre von den quadratischen Resten" (1818) reprinted in Untersuchungen ... pp. 501–505
  7. ^ Lemmermeyer, ex. p. 31, 1.34
  8. ^ Lemmermeyer, pp. 236 ff.

References[edit]

  • Gauss, Carl Friedrich; Maser, H. (translator into German) (1965), Untersuchungen über höhere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition), New York: Chelsea, ISBN 0-8284-0191-8 
  • Gauss, Carl Friedrich; Clarke, Arthur A. (translator into English) (1986), Disquisitiones Arithmeticae (Second, corrected edition), New York: Springer, ISBN 0-387-96254-9 
  • Bach, Eric; Shallit, Jeffrey (1996), Algorithmic Number Theory (Vol I: Efficient Algorithms), Cambridge: The MIT Press, ISBN 0-262-02405-5 
  • Hardy, G. H.; Wright, E. M. (1980), An Introduction to the Theory of Numbers (Fifth edition), Oxford: Oxford University Press, ISBN 978-0-19-853171-5 
  • Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer, ISBN 0-387-97329-X 
  • Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Berlin: Springer, ISBN 3-540-66957-4 
  • Ribenboim, Paulo (1996), The New Book of Prime Number Records, New York: Springer, ISBN 0-387-94457-5 

External links[edit]