ℚ-curves and the Lebesgue–Nagell equation

Abstract

for integers x,q,k,y and n, with k≥0 and n≥3. We extend work of the first and third-named authors by finding all solutions in the cases q=41 and q=97. We do this by constructing a Frey–Hellegouarch ℚ-curve defined over the real quadratic field K=ℚ(q), and using the modular method with multi-Frey techniques.