## Consecutive primes and Legendre symbols

### Volume 190 / 2019

#### Abstract

Let $m$ be any positive integer and let $\delta_1,\delta_2\in\{1,-1\}$. We show that for some constant $C_m \gt 0$ there are infinitely many integers $n \gt 1$ with $p_{n+m}-p_n\le C_m$ such that $$ \biggl(\frac{p_{n+i}}{p_{n+j}}\biggr)=\delta_1 \quad\text{and} \quad\biggl(\frac{p_{n+j}}{p_{n+i}}\biggr)=\delta_2 $$ for all $0\le i \lt j\le m$, 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 $0,1,\ldots,m$.