Quadratic Residue/Examples/7

Example of Quadratic Residues

The set of quadratic residues modulo $7$ is:

$\set {1, 2, 4}$

Proof

To list the quadratic residues of $7$ it is enough to work out the squares $1^2, 2^2, \dotsc, 6^2$ modulo $7$.

\(\ds 1^2\)

\(\equiv\)

\(\ds 1\)

\(\ds \pmod 7\)

\(\ds 2^2\)

\(\equiv\)

\(\ds 4\)

\(\ds \pmod 7\)

\(\ds 3^2\)

\(\equiv\)

\(\ds 2\)

\(\ds \pmod 7\)

\(\ds 4^2\)

\(\equiv\)

\(\ds 2\)

\(\ds \pmod 7\)

\(\ds 5^2\)

\(\equiv\)

\(\ds 4\)

\(\ds \pmod 7\)

\(\ds 6^2\)

\(\equiv\)

\(\ds 1\)

\(\ds \pmod 7\)

So the set of quadratic residues modulo $7$ is:

$\set {1, 2, 4}$

The set of quadratic non-residues of $7$ therefore consists of all the other non-zero least positive residues:

$\set {3, 5, 6}$

$\blacksquare$

Sources

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