Binary Cubic Forms and Rational Cube Sum Problem
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909213727391744 |
|---|---|
| author | Jha, Somnath Majumdar, Dipramit Sury, B. |
| author_facet | Jha, Somnath Majumdar, Dipramit Sury, B. |
| contents | In this note, we use integral binary cubic forms to study the rational cube sum problem. We prove (unconditionally) that for any positive integer $d$, infinitely many primes in each of the residue classes $ 1 \pmod {9d}$ as well as $ -1 \pmod {9d}$, are sums of two rational cubes. Among other results, we prove that every non-zero residue class $a \pmod {q}$, for any prime $q$, contains infinitely many primes which are sums of two rational cubes. Further, for an arbitrary integer $N$, we show there are infinitely many primes $p$ in each of the residue classes $ 8 \pmod 9$ and $1 \pmod 9$, such that $Np$ is a sum of two rational cubes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_06970 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Binary Cubic Forms and Rational Cube Sum Problem Jha, Somnath Majumdar, Dipramit Sury, B. Number Theory 11D25, 11N32, 11N13, 11G05 In this note, we use integral binary cubic forms to study the rational cube sum problem. We prove (unconditionally) that for any positive integer $d$, infinitely many primes in each of the residue classes $ 1 \pmod {9d}$ as well as $ -1 \pmod {9d}$, are sums of two rational cubes. Among other results, we prove that every non-zero residue class $a \pmod {q}$, for any prime $q$, contains infinitely many primes which are sums of two rational cubes. Further, for an arbitrary integer $N$, we show there are infinitely many primes $p$ in each of the residue classes $ 8 \pmod 9$ and $1 \pmod 9$, such that $Np$ is a sum of two rational cubes. |
| title | Binary Cubic Forms and Rational Cube Sum Problem |
| topic | Number Theory 11D25, 11N32, 11N13, 11G05 |
| url | https://arxiv.org/abs/2301.06970 |