Important: The numtheory package 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.

$\mathrm{with}\left(\mathrm{numtheory}\right)\:$

$\mathrm{jacobi}\left(12\,3\right)$

$0$

$\mathrm{jacobi}\left(28\,21\right)$

$0$

$\mathrm{jacobi}\left(6\,11\right)$

$\mathrm{-1}$

$\mathrm{legendre}\left(6\,11\right)$

$\mathrm{-1}$

$\mathrm{jacobi}\left(226\,135\right)$

$1$

$\mathrm{jacobi}\left(26\,35\right)$

$\mathrm{-1}$

$\mathrm{jacobi}\left(-286\,4272943\right)$

$1$

$\mathrm{jacobi}\left(-104\,997\right)$

$\mathrm{-1}$

$\mathrm{jacobi}\left(888\,1999\right)$

$\mathrm{-1}$

$\mathrm{jacobi}\left(a\,b\right)$

$\mathrm{jacobi}\left(a\,b\right)$