In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number p: its value at a (nonzero) quadratic residue mod p is 1 and at a non-quadratic residue (non-residue) is −1. Its value at zero is 0. Relationship between quadratic residues modulo a prime and quadratic residues modulo a prime power [closed] ... then it is a quadratic residue $\bmod p^k$ for all ...

*residues modulo an odd prime p is the partition into quadratic residues and quadratic non-residues if and only if the elements of A and B satisfy certain additive properties, thus providing a purely additive characterization of the set of quadratic residues.*Quadratic residues and quadratic nonresidues Kyle Miller Feb 17, 2017 A number ais called a quadratic residue, modulo p, if it is the square of some other number, modulo p. That is to say, ais a quadratic residue if there is a bsuch that a b2 (mod p). A number is called a quadratic nonresidue if it is not a quadratic residue.1 so that (omitting some details) the squares are precisely the even powers of the generator g. This second argument shows incidentally that the product of two quadratic residues is again a quadratic residue, the product of two quadratic nonresidues is a quadratic residue, and the product of a quadratic residue and a quadratic nonresidue is a