### Further results on derived sequences.

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A sequence is called $k$-automatic if the $n$’th term in the sequence can be generated by a finite state machine, reading $n$ in base $k$ as input. We show that for many multiplicative functions, the sequence ${\left(f\left(n\right)\phantom{\rule{4.0pt}{0ex}}\text{mod}\phantom{\rule{4.0pt}{0ex}}v\right)}_{n\ge 1}$ is not $k$-automatic. Among these multiplicative functions are ${\gamma}_{m}\left(n\right),{\sigma}_{m}\left(n\right),\mu \left(n\right)$ et $\phi \left(n\right)$.

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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