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 k , s and N p ′ k , s , 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 k , s and N p ′ k , s . Exact formulas are given for certain values of k and s under some restrictions. In addition, estimation formulas in other cases are given.
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