A sequence is called -automatic if the ’th term in the sequence can be generated by a finite state machine, reading in base as input. We show that for many multiplicative functions, the sequence is not -automatic. Among these multiplicative functions are et .
We study coprime integer solutions to the equation a³ + b³ⁿ = c² using Galois representations and modular forms. This case represents perhaps the last natural family of generalized Fermat equations descended from spherical cases which is amenable to resolution using the so-called modular method. Our techniques involve an elaborate combination of ingredients, ranging from ℚ-curves and a delicate multi-Frey approach, to appeal to intricate image of inertia arguments.
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