PreprintCC BY 4.0arXiv
Prime pairs along rays of prime indices
Abstract
Let pj be the prime with index j. We prove that the ratios m/n for which pm+pn is a square are dense in ℝ>0. The same is true when |pm−pn| 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.
Related
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BibTeX
@misc{hua2026prime,
title = {{Prime pairs along rays of prime indices}},
author = {Shenghao Hua and Sizhe Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07371},
eprint = {2609.07371},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07371},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Prime pairs along rays of prime indices
AU - Shenghao Hua
AU - Sizhe Xie
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07371
UR - https://arxiv.org/abs/2609.07371
AB - 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 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.
ER -