• Distribution of Residues and Primitive Roots

• # Fulltext

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

1. Central University of Jharkhand, CTI Campus, Ratu-Lohardaga Road, Brambe, Ranchi 835 205, India

• # Proceedings – Mathematical Sciences

Volume 130, 2020
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019