Consecutive primes and Legendre symbols

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 $$\left(\frac{p_{n+i}}{p_{n+j}}\right)={\delta }_1\text{and}\left(\frac{p_{n+j}}{p_{n+i}}\right)={\delta }_2$$ 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 $0,1,\dots ,m$.

12 pages, final published version

References in corpus (1)

• Bounded gaps between primes with a given primitive root