Bounded Gaps between Products of Special Primes
In their breakthrough paper in 2006, Goldston, Graham, Pintz and Yıldırım proved several results about bounded gaps between products of two distinct primes. Frank Thorne expanded on this result, proving bounded gaps in the set of square-free numbers with r prime factors for any r ≥ 2, all of which a...
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| Format: | Article |
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MDPI AG
2018
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| Online Access: | http://hdl.handle.net/1721.1/113354 |
| _version_ | 1826207600178364416 |
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| author | Li, Shiyu Chung, Ping Ngai |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Li, Shiyu Chung, Ping Ngai |
| author_sort | Li, Shiyu |
| collection | MIT |
| description | In their breakthrough paper in 2006, Goldston, Graham, Pintz and Yıldırım proved several results about bounded gaps between products of two distinct primes. Frank Thorne expanded on this result, proving bounded gaps in the set of square-free numbers with r prime factors for any r ≥ 2, all of which are in a given set of primes. His results yield applications to the divisibility of class numbers and the triviality of ranks of elliptic curves. In this paper, we relax the condition on the number of prime factors and prove an analogous result using a modified approach. We then revisit Thorne’s applications and give a better bound in each case. Keywords: bounded prime gaps; square-free numbers; modular elliptic curves |
| first_indexed | 2024-09-23T13:52:05Z |
| format | Article |
| id | mit-1721.1/113354 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T13:52:05Z |
| publishDate | 2018 |
| publisher | MDPI AG |
| record_format | dspace |
| spelling | mit-1721.1/1133542022-10-01T17:39:43Z Bounded Gaps between Products of Special Primes Li, Shiyu Chung, Ping Ngai Massachusetts Institute of Technology. Department of Mathematics Chung, Ping Ngai In their breakthrough paper in 2006, Goldston, Graham, Pintz and Yıldırım proved several results about bounded gaps between products of two distinct primes. Frank Thorne expanded on this result, proving bounded gaps in the set of square-free numbers with r prime factors for any r ≥ 2, all of which are in a given set of primes. His results yield applications to the divisibility of class numbers and the triviality of ranks of elliptic curves. In this paper, we relax the condition on the number of prime factors and prove an analogous result using a modified approach. We then revisit Thorne’s applications and give a better bound in each case. Keywords: bounded prime gaps; square-free numbers; modular elliptic curves 2018-01-30T19:58:30Z 2018-01-30T19:58:30Z 2014-03 2014-02 2018-01-24T21:04:58Z Article http://purl.org/eprint/type/JournalArticle 2227-7390 http://hdl.handle.net/1721.1/113354 Chung, Ping and Li, Shiyu. "Bounded Gaps between Products of Special Primes." Mathematics 2, 1 (March 2014): 37-52 © 2014 The Author(s) http://dx.doi.org/10.3390/math2010037 Mathematics Creative Commons Attribution http://creativecommons.org/licenses/by/4.0/ application/pdf MDPI AG Multidisciplinary Digital Publishing Institute |
| spellingShingle | Li, Shiyu Chung, Ping Ngai Bounded Gaps between Products of Special Primes |
| title | Bounded Gaps between Products of Special Primes |
| title_full | Bounded Gaps between Products of Special Primes |
| title_fullStr | Bounded Gaps between Products of Special Primes |
| title_full_unstemmed | Bounded Gaps between Products of Special Primes |
| title_short | Bounded Gaps between Products of Special Primes |
| title_sort | bounded gaps between products of special primes |
| url | http://hdl.handle.net/1721.1/113354 |
| work_keys_str_mv | AT lishiyu boundedgapsbetweenproductsofspecialprimes AT chungpingngai boundedgapsbetweenproductsofspecialprimes |