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