The QuadraticResidue(a, n) command returns $1$ if a is a quadratic residue modulo n, and returns $\mathrm{-1}$ if a is a quadratic non-residue modulo n.

If there exists an integer $b$ such that $b^2$ is congruent to a modulo n, then a is said to be a quadratic residue modulo n. If there does not exist such a $b$, then a is said to be a quadratic non-residue modulo n.

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

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

$1$

$121\mathbf{mod}22$

$11$

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

$1$

$\mathrm{QuadraticResidue}\left(12\,24\right)$

$1$

$6^2\mathbf{mod}24$

$12$

$\mathrm{QuadraticResidue}\left(3\,7\right)$

$\mathrm{-1}$

$\mathrm{seq}\left(a^2\mathbf{mod}7\,a\=0..6\right)$

$0,1,4,2,2,4,1$

$Q≔\mathrm{Matrix}\left(100\&comma;100\&comma;\left(i\&comma;j\right)\&rarr;\mathrm{`if`}\left(i<j\&comma;-1\&comma;\mathrm{QuadraticResidue}\left(j\&comma;i\right)\right)\right)$

$\mathrm{Statistics}:-\mathrm{HeatMap}\left(Q\&comma;\mathrm{colour}\&equals;\left[\mathrm{white}\&comma;\mathrm{black}\right]\right)$

The NumberTheory[QuadraticResidue] command was introduced in Maple 2016.

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