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 .