Quadratic Residue/Examples/11
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Example of Quadratic Residues
The set of quadratic residues modulo $11$ is:
- $\set {1, 3, 4, 5, 9}$
This sequence is A010375 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
To list the quadratic residues of $11$ it is enough to work out the squares $1^2, 2^2, \ldots, 10^2$ modulo $11$.
\(\ds 1^2\) | \(\equiv\) | \(\ds 1\) | \(\ds \pmod {11}\) | |||||||||||
\(\ds 2^2\) | \(\equiv\) | \(\ds 4\) | \(\ds \pmod {11}\) | |||||||||||
\(\ds 3^2\) | \(\equiv\) | \(\ds 9\) | \(\ds \pmod {11}\) | |||||||||||
\(\ds 4^2\) | \(\equiv\) | \(\ds 5\) | \(\ds \pmod {11}\) | |||||||||||
\(\ds 5^2\) | \(\equiv\) | \(\ds 3\) | \(\ds \pmod {11}\) | |||||||||||
\(\ds 6^2\) | \(\equiv\) | \(\ds 3\) | \(\ds \pmod {11}\) | |||||||||||
\(\ds 7^2\) | \(\equiv\) | \(\ds 5\) | \(\ds \pmod {11}\) | |||||||||||
\(\ds 8^2\) | \(\equiv\) | \(\ds 9\) | \(\ds \pmod {11}\) | |||||||||||
\(\ds 9^2\) | \(\equiv\) | \(\ds 4\) | \(\ds \pmod {11}\) | |||||||||||
\(\ds 10^2\) | \(\equiv\) | \(\ds 1\) | \(\ds \pmod {11}\) |
So the set of quadratic residues modulo $11$ is:
- $\set {1, 3, 4, 5, 9}$
The set of quadratic non-residues of $11$ therefore consists of all the other non-zero least positive residues:
- $\set {2, 6, 7, 8, 10}$
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {3-5}$ The Use of Computers in Number Theory: Exercise $6$
- 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $7$: Patterns in Numbers: Gauss