Quadratic Residue/Examples/5

Example of Quadratic Residues

The set of quadratic residues modulo $5$ is:

$\set {1, 4}$

Proof

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:

$\set {2, 3}$

$\blacksquare$

Sources

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