Quadratic Residue/Examples/7
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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 $6$