A1 Refereed original research article in a scientific journal
On the gaps between consecutive primes
Authors: Sun Yu-Chen, Pan Hao
Publisher: WALTER DE GRUYTER GMBH
Publication year: 2022
Journal: Forum Mathematicum
Journal name in source: FORUM MATHEMATICUM
Journal acronym: FORUM MATH
Number of pages: 14
ISSN: 0933-7741
eISSN: 1435-5337
DOI: https://doi.org/10.1515/forum-2021-0140
Publication's open availability at the time of reporting: No Open Access
Publication channel's open availability : Partially Open Access publication channel
Web address : https://www.degruyter.com/document/doi/10.1515/forum-2021-0140/html
Preprint address: https://arxiv.org/pdf/1802.02470.pdf
Abstract
Let p(n) denote the n-th prime. We prove that, for any m >= 1, there exist infinitely many n such that p(n) - p(n-m) <= C-m for some large constant C-m > 0, andp(n+1) - p(n) >= c(m) log n log log n log log log log n/log log log nfor some small constant c(m) > 0. Furthermore, for any fixed positive integer l, there are many positive integers k with (k, l) = 1 such thatp'(k, l) >= ck . log k log log k log log log log k/log log log kwhere p' (k, l) denotes the least prime of the form kn + l with n >= 1, which improves the previous result of Prachar.
Let p(n) denote the n-th prime. We prove that, for any m >= 1, there exist infinitely many n such that p(n) - p(n-m) <= C-m for some large constant C-m > 0, andp(n+1) - p(n) >= c(m) log n log log n log log log log n/log log log nfor some small constant c(m) > 0. Furthermore, for any fixed positive integer l, there are many positive integers k with (k, l) = 1 such thatp'(k, l) >= ck . log k log log k log log log log k/log log log kwhere p' (k, l) denotes the least prime of the form kn + l with n >= 1, which improves the previous result of Prachar.
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