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.
IŞIK, ERMAN; KARA, YASEMİN; and KARAKURT, EKİN ÖZMAN
"On ternary Diophantine equations of signature $(p,p,2)$ over number fields,"
Turkish Journal of Mathematics: Vol. 44:
4, Article 9.
Available at: https://journals.tubitak.gov.tr/math/vol44/iss4/9