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      https://www.ias.ac.in/article/fulltext/pmsc/123/02/0203-0211

    • Keywords

       

      Quadratic residues; primitive roots; finite fields.

    • Abstract

       

      Given an integer $N\geq 3$, we shall prove that for all primes $p\geq(N-2)^2 4^N$, there exists 𝑥 in $(\mathbb{Z}/p\mathbb{Z})^∗$ such that $x,x+1,\ldots,x+N-1$ are all squares (respectively, non-squares) modulo 𝑝. Similarly, for an integer $N\geq 2$, we prove that for all primes $p\geq \exp(2^{5.54N})$, there exists an element $x\in(\mathbb{Z}/p\mathbb{Z})^∗$ such that $x,x+1,\ldots,x+N-1$ are all generators of $(\mathbb{Z}/p\mathbb{Z})^∗$.

    • Author Affiliations

       

      Jagmohan Tanti1 R Thangadurai2

      1. Central University of Jharkhand, CTI Campus, Ratu-Lohardaga Road, Brambe, Ranchi 835 205, India
      2. 2Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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