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

Authors : Erman IŞIK, Yasemin KARA, Ekin ÖZMAN KARAKURT

Pages : 1197-1211

Doi:10.3906/mat-1911-88

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Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :Let $K$ be a totally real number field with narrow class number one and $O_K$ be its ring of integers. We prove that there is a constant $B_K$ depending only on $K$ such that for any prime exponent $p>B_K$ the Fermat type equation $x^p+y^p=z^2$ with $x,y,z\in O_K$ does not have certain type of solutions. Our main tools in the proof are modularity, level lowering, and image of inertia comparisons.
Keywords : Fermat equation, generalized Fermat equation, S units, modularity