k n |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1 | ||||||||||||||||
3 | 0 | 1 | −1 | ||||||||||||||
5 | 0 | 1 | −1 | −1 | 1 | ||||||||||||
7 | 0 | 1 | 1 | −1 | 1 | −1 | −1 | ||||||||||
9 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | ||||||||
11 | 0 | 1 | −1 | 1 | 1 | 1 | −1 | −1 | −1 | 1 | −1 | ||||||
13 | 0 | 1 | −1 | 1 | 1 | −1 | −1 | −1 | −1 | 1 | 1 | −1 | 1 | ||||
15 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | −1 | 1 | 0 | 0 | −1 | 0 | −1 | −1 | ||
17 | 0 | 1 | 1 | −1 | 1 | −1 | −1 | −1 | 1 | 1 | −1 | −1 | −1 | 1 | −1 | 1 | 1 |
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,[1] 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.
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