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PreprintarXiv licencearXiv

Note on the largest prime factors of consecutive integers

  • Zhiyuan Yang
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Abstract

In this paper, we give an alternative proof of Tao-Teräväinen's results by showing &#{n≤ x:P+(n−1)<P+(n)maxiP+(n+ai)}≫ x, &#{n≤ x:P+(n−1)>P+(n)maxiP+(n+ai)}≫ x, where the ai are at most finite in number, with |ai| = O(1). We also obtain some comparison results for five numbers under the Elliott-Halberstam conjecture for friable integers. For any ε>0, we also show &#{n≤ x:1≤ (P+(n+1))/(P+(n))<1+ε}≫ε x/(log x), which improves a previous result ≫ x(log x)−50 due to S. Kim (2020).

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Cite

BibTeXRIS
BibTeX
@misc{yang2026note,
  title = {{Note on the largest prime factors of consecutive integers}},
  author = {Zhiyuan Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23280},
  eprint = {2610.23280},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23280},
}
RIS
TY  - UNPB
TI  - Note on the largest prime factors of consecutive integers
AU  - Zhiyuan Yang
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23280
UR  - https://arxiv.org/abs/2610.23280
AB  - In this paper, we give an alternative proof of Tao-Teräväinen's results by showing &#{n≤ x:P^+(n−1)<P^+(n)max_iP^+(n+a_i)}≫ x, &#{n≤ x:P^+(n−1)>P^+(n)max_iP^+(n+a_i)}≫ x, where the a_i are at most finite in number, with |a_i| = O(1). We also obtain some comparison results for five numbers under the Elliott-Halberstam conjecture for friable integers. For any ε>0, we also show &#{n≤ x:1≤ (P^+(n+1))/(P^+(n))<1+ε}≫_ε x/(log x), which improves a previous result ≫ x(log x)^−50 due to S. Kim (2020).
ER  -