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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.09892 |
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| _version_ | 1866917060393566208 |
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| author | Jha, Somnath Sahoo, Satyabrat |
| author_facet | Jha, Somnath Sahoo, Satyabrat |
| contents | Let $K$ be a totally real number field and $ \mathcal{O}_K$ be the ring of integers of $K$. This manuscript examines the asymptotic solutions of the Fermat equation of signature $(r, r, p)$, specifically $x^r+y^r=dz^p$ over $K$, where $r,p \geq5$ are rational primes and odd $d\in \mathcal{O}_K \setminus \{0\}$. For a certain class of fields $K$, we first prove that the equation $x^r+y^r=dz^p$ has no asymptotic solution $(a,b,c) \in \mathcal{O}_K^3$ with $2 |c$. Then, we study the asymptotic solutions $(a,b,c) \in \mathcal{O}_K^3$ to the equation $x^5+y^5=dz^p$ with $2 \nmid c$. We use the modular method to prove these results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09892 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic Fermat equation of signature $(r, r, p)$ over totally real fields Jha, Somnath Sahoo, Satyabrat Number Theory Let $K$ be a totally real number field and $ \mathcal{O}_K$ be the ring of integers of $K$. This manuscript examines the asymptotic solutions of the Fermat equation of signature $(r, r, p)$, specifically $x^r+y^r=dz^p$ over $K$, where $r,p \geq5$ are rational primes and odd $d\in \mathcal{O}_K \setminus \{0\}$. For a certain class of fields $K$, we first prove that the equation $x^r+y^r=dz^p$ has no asymptotic solution $(a,b,c) \in \mathcal{O}_K^3$ with $2 |c$. Then, we study the asymptotic solutions $(a,b,c) \in \mathcal{O}_K^3$ to the equation $x^5+y^5=dz^p$ with $2 \nmid c$. We use the modular method to prove these results. |
| title | Asymptotic Fermat equation of signature $(r, r, p)$ over totally real fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2502.09892 |