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The Legendre symbol is a function of and defined as = {(),, (). Legendre's original definition was by means of the explicit formula ... Jacobi symbol calculator This ...
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 algebraic number theory the n-th power residue symbol (for an integer n > 2) is a generalization of the (quadratic) Legendre symbol to n-th powers. These symbols are used in the statement and proof of cubic , quartic , Eisenstein , and related higher [ 1 ] reciprocity laws .
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 ...
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 ...
This interpretation of the Legendre symbol as the sign of a permutation can be extended to the Jacobi symbol ( a n ) , {\displaystyle \left({\frac {a}{n}}\right),} where a and n are relatively prime integers with odd n > 0: a is invertible mod n , so multiplication by a on Z / n Z is a permutation and a generalization of Zolotarev's lemma is ...
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.