Quadratic Residue/Examples/29

Example of Quadratic Residues

The set of quadratic residues modulo $29$ is:

$\set {1, 4, 5, 6, 7, 9, 13, 16, 20, 22, 23, 24, 25, 28}$

This sequence is A010391 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

From Square Modulo n Congruent to Square of Inverse Modulo n, to list the quadratic residues of $29$ it is sufficient to work out the squares $1^2, 2^2, \dotsc, \paren {\dfrac {28} 2}^2$ modulo $29$.

So:

\(\ds 1^2\)

\(\equiv\)

\(\ds 1\)

\(\ds \pmod {29}\)

\(\ds 2^2\)

\(\equiv\)

\(\ds 4\)

\(\ds \pmod {29}\)

\(\ds 3^2\)

\(\equiv\)

\(\ds 9\)

\(\ds \pmod {29}\)

\(\ds 4^2\)

\(\equiv\)

\(\ds 16\)

\(\ds \pmod {29}\)

\(\ds 5^2\)

\(\equiv\)

\(\ds 25\)

\(\ds \pmod {29}\)

\(\ds 6^2\)

\(\equiv\)

\(\ds 7\)

\(\ds \pmod {29}\)

\(\ds 7^2\)

\(\equiv\)

\(\ds 20\)

\(\ds \pmod {29}\)

\(\ds 8^2\)

\(\equiv\)

\(\ds 6\)

\(\ds \pmod {29}\)

\(\ds 9^2\)

\(\equiv\)

\(\ds 23\)

\(\ds \pmod {29}\)

\(\ds 10^2\)

\(\equiv\)

\(\ds 13\)

\(\ds \pmod {29}\)

\(\ds 11^2\)

\(\equiv\)

\(\ds 5\)

\(\ds \pmod {29}\)

\(\ds 12^2\)

\(\equiv\)

\(\ds 28\)

\(\ds \pmod {29}\)

\(\ds 13^2\)

\(\equiv\)

\(\ds 24\)

\(\ds \pmod {29}\)

\(\ds 14^2\)

\(\equiv\)

\(\ds 22\)

\(\ds \pmod {29}\)

So the set of quadratic residues modulo $29$ is:

$\set {1, 4, 5, 6, 7, 9, 13, 16, 20, 22, 23, 24, 25, 28}$

The set of quadratic non-residues of $29$ therefore consists of all the other non-zero least positive residues:

$\set {2, 3, 8, 10, 11, 12, 14, 15, 17, 18, 19, 21, 26, 27}$

This sequence is A028742 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 $6$