Distribution of Residues and Primitive Roots

Proceedings Mathematical Sciences

Jagmohan Tanti¹  R Thangadurai² 

[1] Central University of Jharkhand, CTI Campus, Ratu-Lohardaga Road, Brambe, Ranchi 0, India$$;Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 0, India$$

关键词: Quadratic residues;    primitive roots;    finite fields.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

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

【 授权许可】

Unknown