Prime pairs along rays of prime indices

Let $p_j$ be the prime with index $j$. We prove that the ratios $m\mathrm{/}n$ for which $p_m+p_n$ is a square are dense in ${\mathbb{R}}_{>0}$. The same is true when $\mathrm{\mid }{p}_m-p_n\mathrm{\mid }$ is a square. In every nonempty open interval, almost every index can be used as a numerator and as a denominator. We also give a quantitative lower bound for the number of possible partners.