The arithmetic of consecutive polynomial sequences over finite fields

Abstract

Motivated by a question of van der Poorten about the existence of an infinite chain of prime numbers (with respect to some base), in this paper we advance the study of sequences of consecutive polynomials whose coefficients are chosen consecutively from a sequence in a finite field of odd prime characteristic. We study the arithmetic of such sequences, including bounds for the largest degree of irreducible factors, the number of irreducible factors, as well as for the number of such sequences of fixed length in which all the polynomials are irreducible.

Arithmetic Properties of Integers in Chains and Reflections of g-ary Expansions

“…More precisely, both papers [1,17] study sequences D such that the elements of (1.1) are all primes. Analogues of this question for polynomials over finite fields have been considered by Chou and Cohen [7] and more recently by Gómez-Pérez, Ostafe and Sha [9], which have in fact motivated this work. The rest of our motivation comes from a series of recent striking results about primes with restricted digits [3,5,14,15].…”