  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 $0,1,\ldots,m$.
