# Legendre symbol

ap |
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 ≤ |

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 at a (nonzero) quadratic residue mod *p* is 1 and at a non-quadratic residue (*non-residue*) is −1. Its value at 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.

## Contents

## Definition[edit]

Let be an odd prime number. An integer is a quadratic residue modulo if it is congruent to a perfect square modulo and is a quadratic nonresidue modulo otherwise. The **Legendre symbol** is a function of and 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 *a*R*p*, *a*N*p* 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]} Each row in the following table can be seen to exhibit periodicity, just as described.

## Table of values[edit]

The following is a table of values of Legendre symbol with *p* ≤ 127, *a* ≤ 30, *p* odd prime.

ap |
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 reveals the parity of a non-zero integer mod p. That is, given a generator , if then is a quadratic residue if and only if is even. This also shows that half of the non-zero elements in are quadratic residues.
- If then the fact that
- gives us that is the square root of the quadratic residue .

- The Legendre symbol is periodic in its first (or top) argument: if
*a*≡*b*(mod*p*), then - The Legendre symbol is a completely multiplicative function of its top argument:
- 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 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 for small values of
*a*:- For an odd prime
*p*≠ 3, - For an odd prime
*p*≠ 5,

- For an odd prime
- The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … are defined by the recurrence
*F*_{1}=*F*_{2}= 1,*F*_{n+1}=*F*+_{n}*F*_{n−1}. If*p*is a prime number then

- For example,

- This result comes from the theory of Lucas sequences, which are used in primality testing.
^{[4]}See Wall–Sun–Sun prime.

## 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.

- Gauss introduced the quadratic Gauss sum and used the formula

- in his fourth
^{[5]}and sixth^{[6]}proofs of quadratic reciprocity.

- Kronecker's proof
^{[7]}first establishes that

- Reversing the roles of
*p*and*q*, he obtains the relation between (*p*/*q*) and (*q*/*p*).

- One of Eisenstein's proofs
^{[8]}begins by showing that

- 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]

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