The Distribution Properties of Consecutive Quadratic Residue Sequences

Abstract

We consider any prime number $p$ . Let $k,s$ be two positive integers. We are interested in the arithmetic progressions (sequences) with the common difference $s$ and length $k$ , where the sequence entries are from the set of quadratic residue modulo $p$ or the set of quadratic nonresidue modulo $p$ . The numbers of such sequences are denoted as $N_p\left(k,s\right)$ and $N_p^{\prime }\left(k,s\right)$ , respectively. In this paper, we apply analytic number theory methods, in particular, properties of Legendre’s symbol modulo $p$ and character sums, to study the numbers $N_p\left(k,s\right)$ and $N_p^{\prime }\left(k,s\right)$ . Exact formulas are given for certain values of $k$ and $s$ under some restrictions. In addition, estimation formulas in other cases are given.