Consecutive Quadratic Residues And Quadratic Nonresidue Modulo $p$

We show that for any fixed $\eps>0$, there are numbers $\delta>0$ and $p_0\ge 2$ with the following property: for every prime $p\ge p_0$ and every integer $N$ such that $p^{1/(4\sqrt{e})+\eps}\le N\le p$, the sequence $1,2,...,N$ contains at least $\delta N$ quadratic non-residues modulo $p$. We use this result to obtain strong upper bounds on the sizes of the least quadratic non-residues in Beatty and Piatetski--Shapiro sequences.

Let S be an infinite set of non-empty, finite subsets of the nonnegative integers. If p is an odd prime, let c(p) denote the cardinality of the set {T {\in} S : T {\subseteq} {1,...,p-1} and T is a set of quadratic residues (respectively, non-residues) of p}. When S is constructed in various ways from the set of all arithmetic progressions of nonnegative integers, we determine the sharp asymptotic behavior of c(p) as p {\to} +{\infty}. Generalizations and variations of this a...

Let $q\ne \pm1,v^2$ be a fixed integer, and let $x\geq 1$ be a large number. The least prime number $p \geq3 $ such that $q$ is a primitive root modulo $p$ is conjectured to be $p\ll (\log q)(\log \log q)^3),$ where $\gcd(p,q)=1$. This note proves the existence of small primes $p\ll(\log x)^c$, where $c>0$ is a constant, a close approximation to the conjectured upper bound.

We prove that the average of the $k$-th smallest prime quadratic non-residue modulo a prime approximates the $2k$-th smallest prime.

For a prime $p\ge 5$ let $q_0,q_1,\ldots,q_{(p-3)/2}$ be the quadratic residues modulo $p$ in increasing order. We study two $(p-3)/2$-periodic binary sequences $(d_n)$ and $(t_n)$ defined by $d_n=q_n+q_{n+1}\bmod 2$ and $t_n=1$ if $q_{n+1}=q_n+1$ and $t_n=0$ otherwise, $n=0,1,\ldots,(p-5)/2$. For both sequences we find some sufficient conditions for attaining the maximal linear complexity $(p-3)/2$. Studying the linear complexity of $(d_n)$ was motivated by heuristics of Car...

In this note, we revisit a result of Benli's and Pollack's on the number of small prime $k^{th}$ power residues. The proof is based on their idea of using reciprocity laws, but the argument is simplified and we prove a slightly stronger bound.

Let $p\geq 2$ be a large prime, and let $N\gg ( \log p)^{1+\varepsilon}$. This note proves the existence of primitive roots in the short interval $[M,M+N]$, where $M \geq 2$ is a fixed number, and $ \varepsilon>0$ is a small number. In particular, the least primitive root $g(p)= O\left ((\log p)^{1+\varepsilon} \right)$, and the least prime primitive root $g^*(p)= O\left ((\log p)^{1+\varepsilon} \right)$ unconditionally.

We make many new observations on primitive roots modulo primes. For an odd prime $p$ and an integer $c$, we establish a theorem concerning $\sum_g(\frac{g+c}p)$, where $g$ runs over all the primitive roots modulo $p$ among $1,\ldots,p-1$, and $(\frac{\cdot}p)$ denotes the Legendre symbol. On the basis of our numerical computations, we formulate 35 conjectures involving primitive roots modulo primes. For example, we conjecture that for any prime $p$ there is a primitive root $...

Let $p$ be an odd prime. For any $p$-adic integer $a$ we let $\overline{a}$ denote the unique integer $x$ with $-p/2<x<p/2$ and $x-a$ divisible by $p$. In this paper we study some permutations involving quadratic residues modulo $p$. For instance, we consider the following three sequences. \begin{align*} &A_0: \overline{1^2},\ \overline{2^2},\ \cdots,\ \overline{((p-1)/2)^2},\\ &A_1: \overline{a_1},\ \overline{a_2},\ \cdots,\ \overline{a_{(p-1)/2}},\\ &A_2: \overline{g^2},\ \...

Let $p$ be a sufficiently large prime number, $n$ be a positive odd integer with $n|\,p-1$ and $n>p^\varepsilon $, where $\varepsilon$ is a sufficiently small constant. Let $k(p,\,n)$ denote the least positive integer $k$ such that for $x=-k,\,\dots,\,-1,\,1,\,2,\,\dots,\,k$, the numbers $x^n\pmod p$ yield all the non-zero $n$-th power residues modulo $p$. In this paper, we shall prove $$ k(p,\,n)=O(p^{1-\delta}), $$ which improves a result of S. Chowla and H. London in the c...