AN APPLICATION OF THE MODULAR METHOD AND THE SYMPLECTIC ARGUMENT TO A LEBESGUE–NAGELL EQUATION

We study the generalized Lebesgue–Nagell equation x 2 + 7 2 k + 1 = y n in integers x , y , k , n with k ⩾ 0 and n ⩾ 3 . We use the modular method, the multi–Frey technique and a symplectic argument. We also use a recent implementation of algorithms for solving Thue–Mahler equations by Adela Gherga for the cases n = 3 and 5. The combination of the previous methods allows us to get results for the case 2 ∣ y for the first time.