Mills' constant is irrational
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916811854839808 |
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| author | Saito, Kota |
| author_facet | Saito, Kota |
| contents | Let $ \lfloor x \rfloor $ denote the integer part of $ x $. In 1947, Mills constructed a real number $ ξ> 1 $ such that $\lfloor ξ^{3^k} \rfloor$ is always a prime number for every positive integer $k$. We define Mills' constant as the smallest real number $ξ$ satisfying this property. Determining whether this number is irrational has been a long-standing problem. In this paper, we show that Mills' constant is irrational. Furthermore, we obtain partial results on the transcendency of this number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19461 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mills' constant is irrational Saito, Kota Number Theory 11J72, 11J81 Let $ \lfloor x \rfloor $ denote the integer part of $ x $. In 1947, Mills constructed a real number $ ξ> 1 $ such that $\lfloor ξ^{3^k} \rfloor$ is always a prime number for every positive integer $k$. We define Mills' constant as the smallest real number $ξ$ satisfying this property. Determining whether this number is irrational has been a long-standing problem. In this paper, we show that Mills' constant is irrational. Furthermore, we obtain partial results on the transcendency of this number. |
| title | Mills' constant is irrational |
| topic | Number Theory 11J72, 11J81 |
| url | https://arxiv.org/abs/2404.19461 |