PlanetMath: Jacobi symbol

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Jacobi symbol

(Definition)

The Jacobi symbol is a generalization of the Legendre symbol to all odd positive integers.

Let be an odd positive integer, with prime factorization ${p_1}^{e_1} \cdots {p_k}^{e_k}$ . Let $a \geq 0$ be an integer. The Jacobi symbol $\left(\frac{a}{n}\right)$ is defined to be

$\displaystyle \left(\frac{a}{n}\right) = \prod_{i=1}^k \left(\frac{a}{p_i}\right)^{e_i}$

where $\left(\frac{a}{p_i}\right)$ is the Legendre symbol of

and

A further generalization of the Legendre symbol, due to Kronecker, is the Kronecker symbol.

"Jacobi symbol" is owned by mathwizard. [ owner history (1) ]

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See Also: Legendre symbol, Kronecker symbol

Attachments:: calculating the Jacobi symbol (Algorithm) by mathwizard

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Cross-references: Kronecker symbol, prime factorization, integers, positive, odd, Legendre symbol
There are 10 references to this entry.

This is version 6 of Jacobi symbol, born on 2002-04-22, modified 2004-08-24.
Object id is 2863, canonical name is JacobiSymbol.
Accessed 6578 times total.

Classification:

AMS MSC:	11A07 (Number theory :: Elementary number theory :: Congruences; primitive roots; residue systems)
	11A15 (Number theory :: Elementary number theory :: Power residues, reciprocity)

Pending Errata and Addenda

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