On ternary Diophantine equations of signature (p, p, 2) over number fields

Key Findings & Data:We provethat there is a constant BK depending only on K such that for any prime exponent p > $B_K$ the Fermat type equationmathcal $x^p;+;y^p;=;z^2$ with x, y, z ∈ ${mathcal O}_K$ does not have certain type of solutions.

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Conclusion & Impact:Our main tools in the proof are modularity,level lowering, and image of inertia comparisons.

🔬Design:Research Article

📍Scope:Comprehensive Sample

⚙️Variables:Domain Parameters

Abstract

Let K be a totally real number field with narrow class number one and ${mathcal O}_K$ be its ring of integers. We provethat there is a constant BK depending only on K such that for any prime exponent p > $B_K$ the Fermat type equationmathcal $x^p;+;y^p;=;z^2$ with x, y, z ∈ ${mathcal O}_K$ does not have certain type of solutions. Our main tools in the proof are modularity,level lowering, and image of inertia comparisons.

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Cite This Article

IŞIK, E., KARA, Y., OZMAN, E. (2022). On ternary Diophantine equations of signature (p, p, 2) over number fields. Turkish Journal of Mathematics, Cilt: 44 - Sayı: 4, 1197–1211. https://ajindex.com/articles/607894

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