Integers that are sums of two cubes in the cyclotomic $\mathbb{Z}_p$-extension
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916489938862080 |
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| author | Ray, Anwesh |
| author_facet | Ray, Anwesh |
| contents | Let $n$ be a cubefree natural number and $p\geq 5$ be a prime number. Assume that $n$ is not expressible as a sum of the form $x^3+y^3$, where $x,y\in \mathbb{Q}$. In this note, we study the solutions (or lack thereof) to the equation $n=x^3+y^3$, where $x$ and $y$ belong to the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. As an application, consider the case when $n$ is not a sum of rational cubes. Then, we prove that $n$ cannot be a sum of two cubes in certain large families of prime cyclic extensions of $\mathbb{Q}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17921 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integers that are sums of two cubes in the cyclotomic $\mathbb{Z}_p$-extension Ray, Anwesh Number Theory 11R23 Let $n$ be a cubefree natural number and $p\geq 5$ be a prime number. Assume that $n$ is not expressible as a sum of the form $x^3+y^3$, where $x,y\in \mathbb{Q}$. In this note, we study the solutions (or lack thereof) to the equation $n=x^3+y^3$, where $x$ and $y$ belong to the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. As an application, consider the case when $n$ is not a sum of rational cubes. Then, we prove that $n$ cannot be a sum of two cubes in certain large families of prime cyclic extensions of $\mathbb{Q}$. |
| title | Integers that are sums of two cubes in the cyclotomic $\mathbb{Z}_p$-extension |
| topic | Number Theory 11R23 |
| url | https://arxiv.org/abs/2409.17921 |