PreprintarXiv licencearXiv
The lower bound of shifted primes with large prime factors
Abstract
In this paper, we consider the asymptotic density of Tc(x):=#{p≤ x:P+(p−1)≥ pc}, where P+(n) denote the largest prime factor of n. We show that for x→∞, for any 0<c≤ 1/2, one has mathopliminfx→∞Tc(x)/π(x)&≥max(1−16/5ρ(1/c),1−c+g(c)), where g(c)>0 for 11/32<c≤1/2, and g(c)=0 for 0<c≤11/32. In particular, we have g(1/2)>0.0436. This improves a previous result by Liu-Wu-Xi (2020), who showed mathopliminfx→∞Tc(x)/π(x)&≥max(1−4ρ(1/c),1−c), for 0<c≤ 1/2.
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BibTeX
@misc{yang2026lower,
title = {{The lower bound of shifted primes with large prime factors}},
author = {Zhiyuan Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24347},
eprint = {2609.24347},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24347},
}
RIS
TY - UNPB
TI - The lower bound of shifted primes with large prime factors
AU - Zhiyuan Yang
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24347
UR - https://arxiv.org/abs/2609.24347
AB - In this paper, we consider the asymptotic density of T_c(x):=#{p≤ x:P^+(p−1)≥ p^c}, where P^+(n) denote the largest prime factor of n. We show that for x→∞, for any 0<c≤ 1/2, one has mathopliminf_x→∞T_c(x)/π(x)&≥max(1−16/5ρ(1/c),1−c+g(c)), where g(c)>0 for 11/32<c≤1/2, and g(c)=0 for 0<c≤11/32. In particular, we have g(1/2)>0.0436. This improves a previous result by Liu-Wu-Xi (2020), who showed mathopliminf_x→∞T_c(x)/π(x)&≥max(1−4ρ(1/c),1−c), for 0<c≤ 1/2.
ER -