Residue Class Patterns of Consecutive Primes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909319854817280 |
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| author | Lau, Cheuk Fung |
| author_facet | Lau, Cheuk Fung |
| contents | For $m,q \in \mathbb{N}$, we call an $m$-tuple $(a_1,\ldots,a_m) \in \prod_{i=1}^m (\mathbb{Z}/q\mathbb{Z})^\times$ good if there are infinitely many consecutive primes $p_1,\ldots,p_m$ satisfying $p_i \equiv a_i \pmod{q}$ for all $i$. We show that given any $m$ sufficiently large, $q$ squarefree, and $A \subseteq (\mathbb{Z}/q\mathbb{Z})^\times$ with $|A|=\lfloor 71(\log m)^3 \rfloor$, we can form at least one non-constant good $m$-tuple $(a_1,\ldots,a_m) \in \prod_{i=1}^m A$. Using this, we can provide a lower bound for the number of residue class patterns attainable by consecutive primes, and for $m$ large and $φ(q) \gg (\log m)^{10}$ this improves on the lower bound obtained from direct applications of Shiu (2000) and Dirichlet (1837). The main method is modifying the Maynard-Tao sieve found in Banks, Freiberg, and Maynard (2015), where instead of considering the 2nd moment we considered the $r$-th moment, where $r$ is an integer depending on $m$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12819 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Residue Class Patterns of Consecutive Primes Lau, Cheuk Fung Number Theory 11N05, 11N13 (Primary) 11N36 (Secondary) For $m,q \in \mathbb{N}$, we call an $m$-tuple $(a_1,\ldots,a_m) \in \prod_{i=1}^m (\mathbb{Z}/q\mathbb{Z})^\times$ good if there are infinitely many consecutive primes $p_1,\ldots,p_m$ satisfying $p_i \equiv a_i \pmod{q}$ for all $i$. We show that given any $m$ sufficiently large, $q$ squarefree, and $A \subseteq (\mathbb{Z}/q\mathbb{Z})^\times$ with $|A|=\lfloor 71(\log m)^3 \rfloor$, we can form at least one non-constant good $m$-tuple $(a_1,\ldots,a_m) \in \prod_{i=1}^m A$. Using this, we can provide a lower bound for the number of residue class patterns attainable by consecutive primes, and for $m$ large and $φ(q) \gg (\log m)^{10}$ this improves on the lower bound obtained from direct applications of Shiu (2000) and Dirichlet (1837). The main method is modifying the Maynard-Tao sieve found in Banks, Freiberg, and Maynard (2015), where instead of considering the 2nd moment we considered the $r$-th moment, where $r$ is an integer depending on $m$. |
| title | Residue Class Patterns of Consecutive Primes |
| topic | Number Theory 11N05, 11N13 (Primary) 11N36 (Secondary) |
| url | https://arxiv.org/abs/2409.12819 |