numtheory/jacobi(deprecated) - Help

numtheory

 jacobi
 Jacobi symbol

 Calling Sequence jacobi(a, b)

Parameters

 a - integer b - non-negative integer

Description

 • Important: The numtheory[jacobi] command has been deprecated.  Use the superseding command NumberTheory[KroneckerSymbol], NumberTheory[JacobiSymbol] instead.
 • The function jacobi will compute the Jacobi symbol $J\left(\frac{a}{b}\right)$ of a and b.  If the factorization of b is ${\mathrm{p1}}^{\mathrm{k1}}...{\mathrm{ps}}^{\mathrm{ks}}$, then jacobi(a, b) = legendre(a, p1)^k1 * ... * legendre(a, ps)^ks, where $\mathrm{legendre}\left(a,p\right)$ is the Legendre symbol of a and p.
 • Note that jacobi returns unevaluated if given algebraic arguments not of the types specified above.
 • The command with(numtheory,jacobi) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{numtheory}\right):$
 > $\mathrm{jacobi}\left(12,3\right)$
 ${0}$ (1)
 > $\mathrm{jacobi}\left(28,21\right)$
 ${0}$ (2)
 > $\mathrm{jacobi}\left(6,11\right)$
 ${-}{1}$ (3)
 > $\mathrm{legendre}\left(6,11\right)$
 ${-}{1}$ (4)
 > $\mathrm{jacobi}\left(226,135\right)$
 ${1}$ (5)
 > $\mathrm{jacobi}\left(26,35\right)$
 ${-}{1}$ (6)
 > $\mathrm{jacobi}\left(-286,4272943\right)$
 ${1}$ (7)
 > $\mathrm{jacobi}\left(-104,997\right)$
 ${-}{1}$ (8)
 > $\mathrm{jacobi}\left(888,1999\right)$
 ${-}{1}$ (9)
 > $\mathrm{jacobi}\left(a,b\right)$
 ${\mathrm{numtheory:-jacobi}}{}\left({a}{,}{b}\right)$ (10)