Estimating the Number of Primes In Unusual Domains
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908598988177408 |
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| author | Cai, Johnathan Diehl, Ryan Gasarch, William Kim, Ian Sinha, Rohan |
| author_facet | Cai, Johnathan Diehl, Ryan Gasarch, William Kim, Ian Sinha, Rohan |
| contents | The Prime Number Theorem states that the number of primes in $\{1,\ldots,x\}$, denoted $π(x)$, is approximately $\frac{x}{\ln(x)}$. In this paper, we investigate the distribution of primes for domains other than $\N$. First we look at $A_d=\{ x \colon x\equiv 1 \pmod d\}$. We give a heuristic argument to form a conjecture on the number of {\it congruence monoid primes} in $A_d$ that are $\le x$. We then provide empirical evidence that indicates our conjecture is close but may need some correction. Second, we do similar calculations for the Gaussian Integers. Third, we discuss the difficulty of these types of questions for quadratic extensions of ${\sf Z}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15255 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Estimating the Number of Primes In Unusual Domains Cai, Johnathan Diehl, Ryan Gasarch, William Kim, Ian Sinha, Rohan Number Theory The Prime Number Theorem states that the number of primes in $\{1,\ldots,x\}$, denoted $π(x)$, is approximately $\frac{x}{\ln(x)}$. In this paper, we investigate the distribution of primes for domains other than $\N$. First we look at $A_d=\{ x \colon x\equiv 1 \pmod d\}$. We give a heuristic argument to form a conjecture on the number of {\it congruence monoid primes} in $A_d$ that are $\le x$. We then provide empirical evidence that indicates our conjecture is close but may need some correction. Second, we do similar calculations for the Gaussian Integers. Third, we discuss the difficulty of these types of questions for quadratic extensions of ${\sf Z}$. |
| title | Estimating the Number of Primes In Unusual Domains |
| topic | Number Theory |
| url | https://arxiv.org/abs/2510.15255 |