Infinite prime avoidance

Infinite Sequences and Pattern Avoidance

The study of combinatorics on words dates back at least to the beginning of the 20 century and the work of Axel Thue. Thue was the first to give an example of an infinite word over a three letter alphabet that contains no squares (identical adjacent blocks) xx. This result was eventually used to solve some longstanding open problems in algebra and has remarkable connections to other areas of ma...

Asymptotic Behaviour of Some Infinite Products Involving Prime Numbers

Asymptotic behaviour of some infinite products involving prime numbers

The asymptotic behaviour of some entire functions defined via infinite products is investigated as the parameter z tends to infinity in the sector | arg z| < π. These functions arise in the distributions of the number of prime factors of integers and of the number of irreducible factors of monic polynomials over a finite field. Our approach is based on the Mellin transform with a suitable conto...

Connected Reversible Mealy Automata of Prime Size Cannot Generate Infinite Burnside Groups

The simplest example of an infinite Burnside group arises in the class of automaton groups. However there is no known example of such a group generated by a reversible Mealy automaton. It has been proved that, for a connected automaton of size at most 3, or when the automaton is not bireversible, the generated group cannot be Burnside infinite. In this paper, we extend these results to automata...

Simplicial tree-decompositions of infinite graphs. II. The existence of prime decompositions

In this second of three papers on the same subject we obtain a first characterization of the countable graphs that admit a simplicial tree-decomposition into primes. This paper is a continuation of [ 4 ]. The problem it tackles, the existence of simplicial tree-decompositions into primes, was introduced in [ 4 ], and all the notation used is explained there. To repeat just the most basic defini...