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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 .
The Jacobi symbol is a generalization of the Legendre symbol; the main difference is that the bottom number has to be positive and odd, but does not have to be prime. If it is prime, the two symbols agree. It obeys the same rules of manipulation as the Legendre symbol. In particular
In numerical linear algebra, the Jacobi method (a.k.a. the Jacobi iteration method) is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in. The process is then iterated until it converges.
They can then be standardized into the Jacobi polynomials (,). There are several important subclasses of these: Gegenbauer , Legendre , and two types of Chebyshev . Every Laguerre-like polynomial sequence can have its domain shifted, scaled, and/or reflected so that its interval of orthogonality is [ 0 , ∞ ) {\displaystyle [0,\infty )} , and ...
If is an odd prime, this is equal to the Legendre symbol, and decides whether is a quadratic residue modulo . On the other hand, since the equivalence of a n − 1 2 {\displaystyle a^{\frac {n-1}{2}}} to the Jacobi symbol holds for all odd primes, but not necessarily for composite numbers, calculating both and comparing them can be used as a ...
where () is the Legendre symbol. The Jacobi symbol is a generalisation of the Legendre symbol to ( a n ) {\displaystyle \left({\tfrac {a}{n}}\right)} , where n can be any odd integer. The Jacobi symbol can be computed in time O ((log n )²) using Jacobi's generalization of the law of quadratic reciprocity .
Here ( a / c ) is the Jacobi symbol. This is the famous formula of Carl Friedrich Gauss. For b > 0 the Gauss sums can easily be computed by completing the square in most cases. This fails however in some cases (for example, c even and b odd), which can be computed relatively easy by other means. For example, if c is odd and gcd(a, c) = 1 ...
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x 4 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence x 4 ≡ p (mod q) to that of x 4 ≡ q (mod p).