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Quadratic Residue

If there is an integer such that(1)i.e., the congruence (1) has a solution, then is said to be a quadratic residue (mod ). Note that the trivial case is generally excluded from lists of quadratic residues (e.g., Hardy and Wright 1979, p. 67) so that the number of quadratic residues (mod ) is taken to be one less than the number of squares (mod ). However, other sources include 0 as a quadratic residue.If the congruence does not have a solution, then is said to be a quadratic nonresidue (mod ). Hardy and Wright (1979, pp. 67-68) use the shorthand notations and , to indicated that is a quadratic residue or nonresidue, respectively.In practice, it suffices to restrict the range to , where is the floor function, because of the symmetry .For example, , so 6 is a quadratic residue (mod 10). The entire set of quadratic residues (mod 10) are given by 1, 4, 5, 6, and 9, since (2)(3)(4)making the numbers 2, 3, 7, and 8 the quadratic nonresidues (mod 10).A list..

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