Fermat's Last Theorem over some small real quadratic fields

Using modularity, level lowering, and explicit computations with Hilbert modular forms, Galois representations and ray class groups, we show that for $3 \le d \le 23$squarefree,$d \ne 5$,$17$,theFermatequation$x^n+y^n=z^n$hasnonon-trivialsolutionsoverthequadraticfield$$\mathbb{Q}$(\sqrt{d})$for$n \ge 4$\mathrm{.}Furthermore,weshowfor$d=17$ that the same holds for prime exponents $n\equiv 3$, $5(\mathrm{m}\mathrm{o}\mathrm{d}8)$.