Q-curves and the Lebesgue-Nagell equation

In this paper, we consider the equation x(2) - q(2k+1) = y(n), q inverted iota x, 2 vertical bar y, 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(root q), and using the modular method with multi-Frey techniques.