Quadratic Residue/Examples/3

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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 $3$.

\(\displaystyle 1^2\) \(\equiv\) \(\displaystyle 1\) \(\displaystyle \pmod 3\)
\(\displaystyle 2^2\) \(\equiv\) \(\displaystyle 1\) \(\displaystyle \pmod 3\)


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