Fermat's Last Theorem over some small real quadratic fields

TL;DR

This paper proves that Fermat's Last Theorem holds over certain small real quadratic fields for all exponents n ≥ 4, using advanced modularity and computational techniques, with specific results for d=17 and certain primes.

Contribution

It extends Fermat's Last Theorem to small real quadratic fields using modularity and explicit computations, covering new cases beyond the classical setting.

Findings

No non-trivial solutions for n ≥ 4 over specified quadratic fields

Fermat's Last Theorem holds for d=17 and primes n ≡ 3, 5 mod 8

Uses modularity, level lowering, and Hilbert modular forms

Abstract

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, $d \ne 5$, $17$, the Fermat equation $x^n+y^n=z^n$ has no non-trivial solutions over the quadratic field $\mathbb{Q}(\sqrt{d})$ for $n \ge 4$. Furthermore, we show for $d=17$ that the same holds for prime exponents $n \equiv 3$, $5 \pmod{8}$.