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The Distribution Properties of Consecutive Quadratic Residue Sequences

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  • Xiao Wang
  • Hao Fang
  • Shaofang Hong

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 Npk,s and Np′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 Npk,s and Np′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.

Suggested Citation

  • Xiao Wang & Hao Fang & Shaofang Hong, 2023. "The Distribution Properties of Consecutive Quadratic Residue Sequences," Journal of Mathematics, Hindawi, vol. 2023, pages 1-15, October.
    • Handle: RePEc:hin:jjmath:5253261
    DOI: 10.1155/2023/5253261
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