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:

    No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.
    Florence Nightingale (1820–1910)

    The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.
    William James (1842–1910)

    Scientific method is the way to truth, but it affords, even in
    principle, no unique definition of truth. Any so-called pragmatic
    definition of truth is doomed to failure equally.
    Willard Van Orman Quine (b. 1908)