Quadratic Residue/Examples/5

Example of Quadratic Residues

The set of quadratic residues modulo $5$ is:

$\set {1, 4}$

To list the quadratic residues of $5$ it is enough to work out the squares $1^2, 2^2, 3^2, 4^2$ modulo $5$.

$\ds 1^2$

$\equiv$

$\ds 1$

$\ds \pmod 5$

$\ds 2^2$

$\equiv$

$\ds 4$

$\ds \pmod 5$

$\ds 3^2$

$\equiv$

$\ds 4$

$\ds \pmod 5$

$\ds 4^2$

$\equiv$

$\ds 1$

$\ds \pmod 5$

So the set of quadratic residues modulo $5$ is:

$\set {1, 4}$

The set of quadratic non-residues of $5$ therefore consists of all the other non-zero least positive residues:

$\set {2, 3}$

$\blacksquare$

1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {3-5}$ The Use of Computers in Number Theory: Exercise $6$