Complex Quadratic Residues

17+

real part of modulus

17

imaginary part of modulus

1

plot quadratic residue

1

1 ≡

2

1

mod(17+)

The squares of the numbers from 0 to 14 are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196.

Those squares

(mod15)

are 0, 1, 4, 9, 1, 10, 6, 4, 4, 6, 10, 1, 9, 4, 1. The numbers 0, 1, 4, 6, 9, 10 are the possible quadratic residues

(mod15)

. The missing numbers 2, 3, 5, 7, 8, 11, 12, 13, 14 are the nonresidues.

What happens with a complex modulus? This Demonstration plots complex quadratic residues. The largest value needed to find all nonzero residues

(moda+bi)

is

ceiling((

2

a

+

2

b

)2)

.