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Jacobi symbol
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(Definition)
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The Jacobi symbol is a generalization of the Legendre symbol to all odd positive integers.
Let $n$ 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$$ \left(\frac{a}{n}\right) = \prod_{i=1}^k \left(\frac{a}{p_i}\right)^{e_i}$$ where $\left(\frac{a}{p_i}\right)$ is the Legendre symbol of $a$ and $p_i$ .
A further generalization of the Legendre symbol, due to Kronecker, is the Kronecker symbol.
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"Jacobi symbol" is owned by mathwizard. [ owner history (1) ]
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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 7666 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) |
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Pending Errata and Addenda
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