Gauss Circle Primes
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866929707593760768 |
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| author | Ehrenborg, Thomas |
| author_facet | Ehrenborg, Thomas |
| contents | Given a circle of radius $r$ centered at the origin, the Gauss Circle Problem concerns counting the number of lattice points $C(r)$ within this circle. It is known that as $r$ grows large, the number of lattice points approaches $πr^2$, that is, the area of the circle. The present research is to study how often $C(r)$ will return a prime number of lattice points for $r \leq n$. The Prime Number Theorem predicts that the number of primes less than or equal to $n$ is asymptotic to $\frac{n}{\log n}$. We find that the number of Gauss Circle Primes for $r \leq n$ is also of order $\frac{n}{\log n}$ for $n \leq 2 \times 10^6$. We include a heuristic argument that the Gauss Circle Primes can be approximated by $\frac{n}{\log n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_06804 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gauss Circle Primes Ehrenborg, Thomas General Mathematics Given a circle of radius $r$ centered at the origin, the Gauss Circle Problem concerns counting the number of lattice points $C(r)$ within this circle. It is known that as $r$ grows large, the number of lattice points approaches $πr^2$, that is, the area of the circle. The present research is to study how often $C(r)$ will return a prime number of lattice points for $r \leq n$. The Prime Number Theorem predicts that the number of primes less than or equal to $n$ is asymptotic to $\frac{n}{\log n}$. We find that the number of Gauss Circle Primes for $r \leq n$ is also of order $\frac{n}{\log n}$ for $n \leq 2 \times 10^6$. We include a heuristic argument that the Gauss Circle Primes can be approximated by $\frac{n}{\log n}$. |
| title | Gauss Circle Primes |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2502.06804 |