Quadratic Residue/Examples/61

Example of Quadratic Residues
The set of quadratic residues modulo $61$ is:
 * $\set {1, 3, 4, 5, 9, 12, 13, 14, 15, 16, 19, 20, 22, 25, 27, 34, 36, 39, 41, 42, 45, 46, 47, 48, 49, 53, 56, 57, 58, 60}$

Proof
From Square Modulo n Congruent to Square of Inverse Modulo n, to list the quadratic residues of $61$ it is sufficient to work out the squares $1^2, 2^2, \dotsc, \paren {\dfrac {60} 2}^2$ modulo $61$.

So:

So the set of quadratic residues modulo $61$ is:
 * $\set {1, 3, 4, 5, 9, 12, 13, 14, 15, 16, 19, 20, 22, 25, 27, 34, 36, 39, 41, 42, 45, 46, 47, 48, 49, 53, 56, 57, 58, 60}$

The set of quadratic non-residues of $61$ therefore consists of all the other non-zero least positive residues:
 * $\set {2, 6, 7, 8, 10, 11, 17, 18, 21, 23, 24, 26, 28, 29, 30, 31, 32, 33, 35, 37, 38, 40, 43, 44, 50, 51, 52, 54, 55, 59}$