Quadratic Residue/Examples/3

Example of Quadratic Residues

There exists exactly $1$ quadratic residue modulo $3$, and that is $1$.

Proof

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

\(\ds 1^2\)

\(\equiv\)

\(\ds 1\)

\(\ds \pmod 3\)

\(\ds 2^2\)

\(\equiv\)

\(\ds 1\)

\(\ds \pmod 3\)

So the set of quadratic residues modulo $3$ is:

$\set 1$

The set of quadratic non-residues of $3$ therefore consists of all the other non-zero least positive residues:

$\set 2$

$\blacksquare$

Sources

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