Let n ≥ 2 be an integer. We study the generalized Fermat equation x¹³ + y¹³ = zⁿ in coprime integers x, y, z. Using a combination of techniques, including the modular method, classical descent, unit sieves, and Chabauty and Mordell–Weil sieve methods over number fields, we show that for n = 5 all solutions are trivial. Under the assumption of GRH, we also show that for n = 7 there are only trivial solutions. Furthermore, we provide partial results towards solving the equation for general n, in particular that any solution with 13 dividing z is trivial.