quadratic residuosity of a number
The QuadraticResidue(a, n) command returns 1 if a is a quadratic residue modulo n, and returns −1 if a is a quadratic non-residue modulo n.
If there exists an integer b such that b2 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.
Numbers congruent to a perfect square are always quadratic residues. The converse is true as well.
12 is a quadratic residue modulo 24.
3 is not a quadratic residue modulo 7.
In the following plot, for each row index i and column index j, if the box indexed by i and j is black then j is a quadratic residue modulo i. If the box is white then j is a quadratic non-residue modulo i.
Q≔1−1−1−1−1−1−1−1−1−1…11−1−1−1−1−1−1−1−1…1−11−1−1−1−1−1−1−1…1−1−11−1−1−1−1−1−1…1−1−111−1−1−1−1−1…1−111−11−1−1−1−1…11−11−1−11−1−1−1…1−1−11−1−1−11−1−1…1−1−11−1−11−11−1…1−1−1111−1−111…⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮100 × 100 Matrix
The NumberTheory[QuadraticResidue] command was introduced in Maple 2016.
For more information on Maple 2016 changes, see Updates in Maple 2016.
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