How often are $ \lfloor {n^α} \rfloor $ and $ \lfloor {n^β} \rfloor $ simultaneously primes?
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917159963197440 |
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| author | Dixit, Anup B. Kumar, Nikhil S |
| author_facet | Dixit, Anup B. Kumar, Nikhil S |
| contents | Let $ \lfloor {x} \rfloor $ denote the greatest integer less than or equal to a real number $x$. Given real numbers $0<α_1 < α_2 < \cdots< α_k < 1$ satisfying a certain condition, we show that there are infinitely many positive integers $n$ for which all of $ \lfloor{n^{α_1}}\rfloor, \lfloor{n^{α_2}}\rfloor,\ldots, \lfloor{n^{α_k}}\rfloor $ are prime numbers. Our approach relies on establishing a simultaneous equidistribution theorem for $ \lfloor{n^{α_i}}\rfloor $ across $k$-many arithmetic progressions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18233 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | How often are $ \lfloor {n^α} \rfloor $ and $ \lfloor {n^β} \rfloor $ simultaneously primes? Dixit, Anup B. Kumar, Nikhil S Number Theory Let $ \lfloor {x} \rfloor $ denote the greatest integer less than or equal to a real number $x$. Given real numbers $0<α_1 < α_2 < \cdots< α_k < 1$ satisfying a certain condition, we show that there are infinitely many positive integers $n$ for which all of $ \lfloor{n^{α_1}}\rfloor, \lfloor{n^{α_2}}\rfloor,\ldots, \lfloor{n^{α_k}}\rfloor $ are prime numbers. Our approach relies on establishing a simultaneous equidistribution theorem for $ \lfloor{n^{α_i}}\rfloor $ across $k$-many arithmetic progressions. |
| title | How often are $ \lfloor {n^α} \rfloor $ and $ \lfloor {n^β} \rfloor $ simultaneously primes? |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.18233 |