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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.
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.
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. Example. Taking p = 11 and a = 7, the relevant sequence of integers is 7, 14, 21, 28, 35. After reduction modulo 11, this sequence ...
Gauss's third proof of quadratic reciprocity, as modified by Eisenstein, has two basic steps. [20] [21] Let p and q be distinct positive odd prime numbers, and let = (), = (). First, Gauss's lemma is used to show that the Legendre symbols are given by
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections ...
In number theory, a symbol is any of many different generalizations of the Legendre symbol.This article describes the relations between these various generalizations. The symbols below are arranged roughly in order of the date they were introduced, which is usually (but not always) in order of increasing generality.
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