Quadratic Residue/Examples/17

Example of Quadratic Residues

The set of quadratic residues modulo $17$ is:

$\set {1, 2, 4, 8, 9, 13, 15, 16}$

This sequence is A010379 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

To list the quadratic residues of $17$ it is enough to work out the squares $1^2, 2^2, \dotsc, 16^2$ modulo $17$.

\(\ds 1^2\)

\(\equiv\)

\(\ds 1\)

\(\ds \pmod {17}\)

\(\ds 2^2\)

\(\equiv\)

\(\ds 4\)

\(\ds \pmod {17}\)

\(\ds 3^2\)

\(\equiv\)

\(\ds 9\)

\(\ds \pmod {17}\)

\(\ds 4^2\)

\(\equiv\)

\(\ds 16\)

\(\ds \pmod {17}\)

\(\ds 5^2\)

\(\equiv\)

\(\ds 8\)

\(\ds \pmod {17}\)

\(\ds 6^2\)

\(\equiv\)

\(\ds 2\)

\(\ds \pmod {17}\)

\(\ds 7^2\)

\(\equiv\)

\(\ds 15\)

\(\ds \pmod {17}\)

\(\ds 8^2\)

\(\equiv\)

\(\ds 13\)

\(\ds \pmod {17}\)

\(\ds 9^2\)

\(\equiv\)

\(\ds 13\)

\(\ds \pmod {17}\)

\(\ds 10^2\)

\(\equiv\)

\(\ds 15\)

\(\ds \pmod {17}\)

\(\ds 11^2\)

\(\equiv\)

\(\ds 2\)

\(\ds \pmod {17}\)

\(\ds 12^2\)

\(\equiv\)

\(\ds 8\)

\(\ds \pmod {17}\)

\(\ds 13^2\)

\(\equiv\)

\(\ds 16\)

\(\ds \pmod {17}\)

\(\ds 14^2\)

\(\equiv\)

\(\ds 9\)

\(\ds \pmod {17}\)

\(\ds 15^2\)

\(\equiv\)

\(\ds 4\)

\(\ds \pmod {17}\)

\(\ds 16^2\)

\(\equiv\)

\(\ds 1\)

\(\ds \pmod {17}\)

So the set of quadratic residues modulo $17$ is:

$\set {1, 2, 4, 8, 9, 13, 15, 16}$

The set of quadratic non-residues of $17$ therefore consists of all the other non-zero least positive residues:

$\set {3, 5, 6, 7, 10, 11, 12, 14}$

This sequence is A028730 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

$\blacksquare$

Sources

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