On the generalized Fermat equation x¹³ + y¹³ = zⁿ

Let n ∈ Z≥ 2. We study the generalized Fermat equation \[x¹³+y¹³=z^n, x,y,z ∈ Z, (x,y,z)=1.\] 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 its solutions $(a,b,c)$ are trivial, i.e. satisfy $abc=0$. 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 ∈ Z≥ 2, in particular that any solution $(a,b,c)$ with 13 c is trivial.