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:

$\set {2, 3}$

$\blacksquare$