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Multiplicative functions and k -automatic sequences

Soroosh Yazdani — 2001

Journal de théorie des nombres de Bordeaux

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 ( f ( n ) mod v ) n ≥ 1 is not k -automatic. Among these multiplicative functions are γ m ( n ) , σ m ( n ) , μ ( n ) et φ ( n ) .

On the equation a³ + b³ⁿ = c²

Michael A. Bennett;  Imin Chen;  Sander R. Dahmen;  Soroosh Yazdani — 2014

Acta Arithmetica

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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