A309680 The smallest nonsquare nonzero integer that is a quadratic residue modulo n, or 0 if no such integer exists.

0, 0, 0, 0, 0, 3, 2, 0, 7, 5, 3, 0, 3, 2, 6, 0, 2, 7, 5, 5, 7, 3, 2, 12, 6, 3, 7, 8, 5, 6, 2, 17, 3, 2, 11, 13, 3, 5, 3, 20, 2, 7, 6, 5, 10, 2, 2, 33, 2, 6, 13, 12, 6, 7, 5, 8, 6, 5, 3, 21, 3, 2, 7, 17, 10, 3, 6, 8, 3, 11, 2, 28, 2, 3, 6, 5, 11, 3, 2, 20, 7

Offset: 1

Author

John Prosser, Aug 12 2019

Keywords

• nonn

Examples

			For n=5, the nonzero quadratic residues modulo 5 are 1 and 4. Since these are both squares we have a(5) = 0.
For n=6, the nonzero quadratic residues modulo 6 are 1,3, and 4. Since 3 is not a square we have a(6) = 3.
For n=10, the nonzero quadratic residues modulo 10 are 1,4,5,6,9. Since 5 is the least nonsquare value, we have a(10) = 5.
		

Crossrefs

A330404 is an alternate version.

Cf. A046071, A053760, A057126.

Programs

Mathematica

a[n_] := SelectFirst[ Union@ Mod[Range[n-1]^2, n], ! IntegerQ@ Sqrt@ # &, 0]; Array[a, 81] (* Giovanni Resta, Aug 13 2019 *)

PARI

a(n) = my(v=select(x->issquare(x), vector(n-1, k, Mod(k,n)), 1), y = select(x->!issquare(x), Vec(v))); if (#y, y[1], 0); \\ Michel Marcus, Aug 16 2019

Formula

a(n) = 2 for n in A057126 and n > 2. - Michel Marcus, Aug 24 2019