The Quadratic Residues and Some of Their New Distribution Properties

In this paper, we give some interesting identities and asymptotic formulas for one kind of counting function, by studying the computational problems involving the symmetry sums of one kind quadratic residues and quadratic non-residues mod p . The main methods we used are the properties of the Legendre’s symbol mod p , and the estimate for character sums. As application, we solve two open problems proposed by Zhiwei Sun.