Turkish Journal of Mathematics
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.
DOI
10.3906/mat-1911-88
Keywords
Fermat equation, generalized Fermat equation, S-units, modularity
First Page
1197
Last Page
1211
Recommended Citation
IŞIK, E, KARA, Y, & KARAKURT, E. Ö (2020). On ternary Diophantine equations of signature $(p,p,2)$ over number fields. Turkish Journal of Mathematics 44 (4): 1197-1211. https://doi.org/10.3906/mat-1911-88
