Jacobi Symbol - Definition

Definition

For any integer a and any positive odd integer n the Jacobi symbol is defined as the product of the Legendre symbols corresponding to the prime factors of n:


represents the Legendre symbol, defined for all integers a and all odd primes p by

\left(\frac{a}{p}\right) = \begin{cases} \;\;\,0\mbox{ if } a \equiv 0 \pmod{p} \\+1\mbox{ if }a \not\equiv 0\pmod{p} \mbox{ and for some integer }x, \;a\equiv x^2\pmod{p} \\-1\mbox{ if there is no such } x. \end{cases}

Following the normal convention for the empty product, The Legendre and Jacobi symbols are indistinguishable exactly when the lower argument is an odd prime, in which case they have the same value.

Read more about this topic:  Jacobi Symbol

Famous quotes containing the word definition:

    “It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
    —Jane Adams (20th century)

    “Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
    —Nadine Gordimer (b. 1923)

    “The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
    —Jean Baudrillard (b. 1929)