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arXiv · 2609.07371

Prime pairs along rays of prime indices

Shenghao Hua·Sizhe Xie
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Abstract

Let $p_j$ be the prime with index $j$. We prove that the ratios $m/n$ for which $p_m+p_n$ is a square are dense in $\mathbb R_{>0}$. The same is true when $|p_m-p_n|$ 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.

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Shenghao Hua, Sizhe Xie. 2026-09-07. Prime pairs along rays of prime indices. https://arxiv.org/abs/2609.07371

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