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Jacobi symbol

Jacobi symbol k/n for various k (along top) and n (along left side). Only 0 ≤ k < n are shown, since due to rule (2) below any other k can be reduced modulo n. Quadratic residues are highlighted in yellow — note that no entry with a Jacobi symbol of −1 is a quadratic residue, and if k is a quadratic residue modulo a coprime n, then k/n = 1, but not all entries with a Jacobi symbol of 1 (see the n = 9 and n = 15 rows) are quadratic residues. Notice also that when either n or k is a square, all values are nonnegative. The Jacobi symbol is a generalization of the Legendre symbol. Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other branches of number theory, but its main use is in computational number theory, especially primality testing and integer factorization; these in turn are important in cryptography. For any integer a and any positive odd integer n, the Jacobi symbol a/n is defined as the product of the Legendre symbols corresponding to the prime factors of n: where is the prime factorization of n. The Legendre symbol a/p is defined for all integers a and all odd primes p by Following the normal convention for the empty product, a/1 = 1. When the lower argument is an odd prime, the Jacobi symbol is equal to the Legendre symbol. The following is a table of values of Jacobi symbol k/n with n ≤ 59, k ≤ 30, n odd. The following facts, even the reciprocity laws, are straightforward deductions from the definition of the Jacobi symbol and the corresponding properties of the Legendre symbol. The Jacobi symbol is defined only when the upper argument ("numerator") is an integer and the lower argument ("denominator") is a positive odd integer.

  1. If n is (an odd) prime, then the Jacobi symbol a/n is equal to (and written the same as) the corresponding Legendre symbol.
  2. If a ≡ b (mod n), then

If either the top or bottom argument is fixed, the Jacobi symbol is a completely multiplicative function in the remaining argument: 4. 5. The law of quadratic reciprocity: if m and n are odd positive coprime integers, then 6.

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Related concepts (12)
Quadratic residue
In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that: Otherwise, q is called a quadratic nonresidue modulo n. Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.
Legendre symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo of 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 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.
Euler's criterion
In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let p be an odd prime and a be an integer coprime to p. Then Euler's criterion can be concisely reformulated using the Legendre symbol: The criterion first appeared in a 1748 paper by Leonhard Euler. The proof uses the fact that the residue classes modulo a prime number are a field. See the article prime field for more details.
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