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NumberTheory

  

QuadraticResidue

  

quadratic residuosity of a number

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

QuadraticResidue(a, n)

Parameters

a

-

integer

n

-

positive integer

Description

• 

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.

Examples

withNumberTheory:

Numbers congruent to a perfect square are always quadratic residues. The converse is true as well.

QuadraticResidue11,22

1

(1)

121mod22

11

(2)

QuadraticResidue22,11

1

(3)

12 is a quadratic residue modulo 24.

QuadraticResidue12,24

1

(4)

62mod24

12

(5)

3 is not a quadratic residue modulo 7.

QuadraticResidue3,7

−1

(6)

seqa2mod7,a=0..6

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

(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.

QMatrix100&comma;100&comma;i&comma;j&rarr;`if`i<j&comma;1&comma;QuadraticResiduej&comma;i

(8)

Statistics:-HeatMapQ&comma;colour&equals;white&comma;black

Compatibility

• 

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

• 

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

See Also

NumberTheory