The generalized Fermat equation x³ + y³ = zⁿ

We prove that the equations x³ + y³ = zⁿ and x³ + y³ = 3zⁿ have no solution in coprime nonzero integers for any n ≥ 3. For the first equation the open cases were prime exponents p ≡ 1 (mod 3) above 10⁹ outside a set of congruence classes treated by Chen and Siksek, where the solution 1³ + 2³ = 3² blocks the modular method. We replace Kraus's Frey curve by a hypergeometric motive of rank three over K = Q(√−3). A solution gives a parameter 3-adically close to the point of maximally unipotent monodromy, where inertia at √−3 acts unipotently on the fibers. The parameter of 1³ + 2³ = 3² is farther away. By a theorem of Calegari, Emerton and Gee, the mod p representation of the fiber is the reduction of a member of a compatible system, whose member at a prime above 3 is irreducible and ordinary at √−3. A ramification computation at √−3, unconditional discriminant bounds and finite group theory show that every reduction of this member has trivial semisimplification. As the maximal pro-3 extension of K unramified outside √−3 is generated by one inertia group, the member stabilizes its ordinary line, a contradiction. Frey curves, descents and the same argument, extended to the prime exponents p ≡ 2 (mod 3), which are inert in K, settle x³ + y³ = 3zⁿ.