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