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arXiv · 2502.09892

Asymptotic Fermat equation of signature $(r, r, p)$ over totally real fields

Somnath Jha·Satyabrat Sahoo
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Abstract

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.

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Somnath Jha, Satyabrat Sahoo. 2025-02-14. Asymptotic Fermat equation of signature $(r, r, p)$ over totally real fields. https://doi.org/10.1007/s11139-025-01168-6

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