Existence of primes in the interval $ [15x,16x] $ -- An entirely elementary proof --
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912517872156672 |
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| author | Aoki, Hiroki Higa, Riku Sugawara, Ryosei |
| author_facet | Aoki, Hiroki Higa, Riku Sugawara, Ryosei |
| contents | In this paper, we give a short and entirely elementary proof of the proposition ``For any positive integer $ N $, there exists a real number $ L $ such that for any real number $ x \geqq L $, there are at least $ N $ primes in the interval $ [kx, (k+1)x] $'' for $ k \leqq 15 $. Our proof is based on the idea of the proof by Erdös for $ k=1 $ and its improvement by Hitotsumatsu and by Sainose for $ k=2 $. In the case of $ k=3 $ and $ k=4 $, the method is very similar to the case of $ k=2 $, however, in the case of $ k \geqq 5 $, we need new idea to complete the proof. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06069 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of primes in the interval $ [15x,16x] $ -- An entirely elementary proof -- Aoki, Hiroki Higa, Riku Sugawara, Ryosei Number Theory 11A41, 11N05 In this paper, we give a short and entirely elementary proof of the proposition ``For any positive integer $ N $, there exists a real number $ L $ such that for any real number $ x \geqq L $, there are at least $ N $ primes in the interval $ [kx, (k+1)x] $'' for $ k \leqq 15 $. Our proof is based on the idea of the proof by Erdös for $ k=1 $ and its improvement by Hitotsumatsu and by Sainose for $ k=2 $. In the case of $ k=3 $ and $ k=4 $, the method is very similar to the case of $ k=2 $, however, in the case of $ k \geqq 5 $, we need new idea to complete the proof. |
| title | Existence of primes in the interval $ [15x,16x] $ -- An entirely elementary proof -- |
| topic | Number Theory 11A41, 11N05 |
| url | https://arxiv.org/abs/2503.06069 |