PreprintCC BY 4.0arXiv
A note on large clusters of E2-numbers
Abstract
Let qn denote the nth product of two distinct primes. By completing a square in Sono's sieve and solving the resulting one-dimensional variational problem, we prove unconditionally that, for every ε>0 and all sufficiently large integers ρ, liminfn→∞(qn+ρ−qn) ≤ exp((π+ε)√ρ).
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BibTeX
@misc{zhao2026note,
title = {{A note on large clusters of $E_2$-numbers}},
author = {Genheng Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35548},
eprint = {2609.35548},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35548},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A note on large clusters of E_2-numbers
AU - Genheng Zhao
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35548
UR - https://arxiv.org/abs/2609.35548
AB - Let q_n denote the nth product of two distinct primes. By completing a square in Sono's sieve and solving the resulting one-dimensional variational problem, we prove unconditionally that, for every ε>0 and all sufficiently large integers ρ, liminf_n→∞(q_n+ρ−q_n) ≤ exp((π+ε)√ρ).
ER -