Large gaps between consecutive prime numbers
Let G(X) denote the size of the largest gap between consecutive primes below X. Answering a question of Erdős, we show that $G(X)\geqslant f(X)\frac{logX log logX log log log log X}{(log log logX)^2}$, where f(X) is a function tending to infinity with X. Our proof combines existing arguments with a...
| Main Authors: | , , , |
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| Format: | Journal article |
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Princeton University, Department of Mathematics
2016
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| _version_ | 1797078882709405696 |
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| author | Ford, K Green, B Konyagin, S Tao, T |
| author_facet | Ford, K Green, B Konyagin, S Tao, T |
| author_sort | Ford, K |
| collection | OXFORD |
| description | Let G(X) denote the size of the largest gap between consecutive primes below X. Answering a question of Erdős, we show that $G(X)\geqslant f(X)\frac{logX log logX log log log log X}{(log log logX)^2}$, where f(X) is a function tending to infinity with X. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes. |
| first_indexed | 2024-03-07T00:37:54Z |
| format | Journal article |
| id | oxford-uuid:820abcef-02c4-4f57-85f7-25a5575bcabe |
| institution | University of Oxford |
| last_indexed | 2024-03-07T00:37:54Z |
| publishDate | 2016 |
| publisher | Princeton University, Department of Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:820abcef-02c4-4f57-85f7-25a5575bcabe2022-03-26T21:34:34ZLarge gaps between consecutive prime numbersJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:820abcef-02c4-4f57-85f7-25a5575bcabeSymplectic Elements at OxfordPrinceton University, Department of Mathematics2016Ford, KGreen, BKonyagin, STao, TLet G(X) denote the size of the largest gap between consecutive primes below X. Answering a question of Erdős, we show that $G(X)\geqslant f(X)\frac{logX log logX log log log log X}{(log log logX)^2}$, where f(X) is a function tending to infinity with X. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes. |
| spellingShingle | Ford, K Green, B Konyagin, S Tao, T Large gaps between consecutive prime numbers |
| title | Large gaps between consecutive prime numbers |
| title_full | Large gaps between consecutive prime numbers |
| title_fullStr | Large gaps between consecutive prime numbers |
| title_full_unstemmed | Large gaps between consecutive prime numbers |
| title_short | Large gaps between consecutive prime numbers |
| title_sort | large gaps between consecutive prime numbers |
| work_keys_str_mv | AT fordk largegapsbetweenconsecutiveprimenumbers AT greenb largegapsbetweenconsecutiveprimenumbers AT konyagins largegapsbetweenconsecutiveprimenumbers AT taot largegapsbetweenconsecutiveprimenumbers |