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If there is no integer such that

i.e., if the congruence (35) has no solution, then is said to be a quadratic nonresidue (mod ). If the congruence (35) does have a solution, then is said to be a quadratic residue (mod ).

In practice, it suffices to restrict the range to , where is the floor function, because of the symmetry .

The following table summarizes the quadratic nonresidues for small (OEIS A105640).

 quadratic nonresidues 1 (none) 2 (none) 3 2 4 2, 3 5 2, 3 6 2, 5 7 3, 5, 6 8 2, 3, 5, 6, 7 9 2, 3, 5, 6, 8 10 2, 3, 7, 8 11 2, 6, 7, 8, 10 12 2, 3, 5, 6, 7, 8, 10, 11 13 2, 5, 6, 7, 8, 11 14 3, 5, 6, 10, 12, 13 15 2, 3, 5, 7, 8, 11, 12, 13, 14 16 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15 17 3, 5, 6, 7, 10, 11, 12, 14 18 2, 3, 5, 6, 8, 11, 12, 14, 15, 17 19 2, 3, 8, 10, 12, 13, 14, 15, 18 20 2, 3, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19

The numbers of quadratic nonresidues (mod ) for , 2, ... are 0, 0, 1, 2, 2, 2, 3, 5, 5, 4, 5, 8, 6, 6, ... (OEIS A095972).

The smallest quadratic nonresidues for , 4, ... are 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, ... (OEIS A020649). The smallest quadratic nonresidues for , 3, 5, 7, 11, ... are 2, 2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, 3, ... (OEIS A053760).

If the extended Riemann hypothesis is true, then the first quadratic nonresidue of a number (mod ) is always less than (Wedeniwski 2001) for .

The following table gives the values of such that the least quadratic nonresidue is for small .

 OEIS such that is the smallest quadratic nonresidue 2 A025020 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, ... 3 A025021 7, 14, 17, 31, 34, 41, 49, 62, 79, 82, ... 5 A025022 23, 46, 47, 73, 94, 97, 146, 167, 193, ... 7 A025023 71, 142, 191, 239, 241, 359, 382, ...

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## References

Sloane, N. J. A. Sequences A020649, A025020, A025021, A025022, A025023, A053760, A095972, and A105640 in "The On-Line Encyclopedia of Integer Sequences."Wedeniwski, S. "Primality Tests on Commutator Curves." Dissertation. Tübingen, Germany, 2001. http://www.hipilib.de/prime/primality-tests-on-commutator-curves.pdf.