The KroneckerSymbol(a, n) command computes the Kronecker symbol of a and n.

The alternative calling sequences, JacobiSymbol(a, m) and LegendreSymbol(a, m), have return values equal to KroneckerSymbol(a, m), but m must be a positive odd integer.

The Legendre symbol is typically defined only for second arguments that are prime, but due to primality checking being expensive, here LegendreSymbol is an alias of JacobiSymbol.

If n is equal to $u\left(\prod _{i=1}^k{p}_i^{d_i}\right)$ where $u$ is $\mathrm{-1}$ or $1$ and the $p_i$ are distinct primes, then the Kronecker symbol $\left(\frac{a}{n}\right)$ is given by $\left(\frac{a}{u}\right)\prod _{i=1}^k{\left(\frac{a}{p_i}\right)}^{d_i}$, where $\left(\frac{a}{k}\right)$ is the usual Legendre symbol, except for the following cases.

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

$\mathrm{LegendreSymbol}\left(22\,11\right)$

$0$

$\mathrm{LegendreSymbol}\left(9\,11\right)$

$1$

$\mathrm{LegendreSymbol}\left(10\,11\right)$

$\mathrm{-1}$

$\mathrm{KroneckerSymbol}\left(11\,3^2\right)$

$1$

$\mathrm{KroneckerSymbol}\left(9\,2\cdot 11\right)$

$1$

The NumberTheory[KroneckerSymbol], NumberTheory[JacobiSymbol] and NumberTheory[LegendreSymbol] commands were introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.