Remarks on the distribution of the sum of k quadratic residues (non-residues) modulo a prime

Let [Formula: see text] be an odd prime, [Formula: see text] be the set of quadratic residues modulo [Formula: see text], and [Formula: see text] be the set of quadratic non-residues modulo [Formula: see text]. In this paper, we observe that the problem of finding the number of solutions of [Formula: see text] over [Formula: see text], where [Formula: see text] is equivalent to the problem of counting the multiplicity (see Definition 1) of [Formula: see text], [Formula: see text], or zero in the set of sums of [Formula: see text] quadratic residues, which are not necessarily distinct, depending on whether [Formula: see text] is a quadratic residue, a non-residue, or zero, respectively. Thus, we count the multiplicity of [Formula: see text], [Formula: see text], or zero in the set of sums of [Formula: see text] quadratic residues using a graphical approach. In addition, we compute the number of times the sum of [Formula: see text] squares and [Formula: see text] non-squares over [Formula: see text] yields a square, non-square, or zero. Furthermore, we obtain an explicit formula for the sum of [Formula: see text] distinct quadratic residues (non-residues) modulo [Formula: see text], where [Formula: see text]. The results obtained in this study are useful in the factorization of the polynomial [Formula: see text] over [Formula: see text], where [Formula: see text] is a quadratic residue modulo [Formula: see text], and are also helpful in developing time-efficient computer programs.