Consecutive primes and Legendre symbols

TL;DR

This paper proves the existence of infinitely many prime tuples with prescribed Legendre symbol patterns and, under GRH, prime tuples where certain primes are primitive roots modulo others, revealing deep properties of prime distributions.

Contribution

It establishes new results on the distribution of primes with specific Legendre symbol configurations and primitive root relationships, extending understanding of prime patterns under certain conditions.

Findings

Infinitely many prime tuples with fixed Legendre symbol patterns exist.

Under GRH, primes in tuples can be primitive roots modulo each other.

Bounded gaps between primes with these properties are proven.

Abstract

Let $m$ be any positive integer and let $\delta_1,\delta_2\in\{1,-1\}$. We show that for some constanst $C_m>0$ there are infinitely many integers $n>1$ with $p_{n+m}-p_n\le C_m$ such that $$\left(\frac{p_{n+i}}{p_{n+j}}\right)=\delta_1\ \quad\text{and}\ \quad\left(\frac{p_{n+j}}{p_{n+i}}\right)=\delta_2$$ for all $0\le i<j\le m$, where $p_k$ denotes the $k$-th prime, and $(\frac {\cdot}p)$ denotes the Legendre symbol for any odd prime $p$. We also prove that under the Generalized Riemann Hypothesis there are infinitely many positive integers $n$ such that $p_{n+i}$ is a primitive root modulo $p_{n+j}$ for any distinct $i$ and $j$ among $0,1,\ldots,m$.