Quadratic Residue/Examples/7

Example of Quadratic Residues

The set of quadratic residues modulo $7$ is:

$\set {1, 2, 4}$

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:

$\set {3, 5, 6}$

$\blacksquare$

1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {3-5}$ The Use of Computers in Number Theory: Exercise $6$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): residue: 2.
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): residue: 2.