PreprintCC BY 4.0arXiv
On the largest prime factors of p−1 and p+1
Abstract
We prove that each of the inequalities P+(p+1)>P+(p−1) and P+(p−1)>P+(p+1) holds for a positive proportion of primes. The argument combines a logarithmic balance identity with a uniform Selberg sieve over a thin cofactor strip. The same transfer mechanism applies to either shift and converts a one-sided large-prime-factor reservoir into an unconditional ordering result in both directions.
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BibTeX
@misc{zhao2026largest,
title = {{On the largest prime factors of $p-1$ and $p+1$}},
author = {Genheng Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.35610},
eprint = {2609.35610},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35610},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the largest prime factors of p−1 and p+1
AU - Genheng Zhao
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.35610
UR - https://arxiv.org/abs/2609.35610
AB - We prove that each of the inequalities P^+(p+1)>P^+(p−1) and P^+(p−1)>P^+(p+1) holds for a positive proportion of primes. The argument combines a logarithmic balance identity with a uniform Selberg sieve over a thin cofactor strip. The same transfer mechanism applies to either shift and converts a one-sided large-prime-factor reservoir into an unconditional ordering result in both directions.
ER -