Local structure of classical sequences, regular sequences, and dynamics

TL;DR

This paper introduces new notions of local and algebraic realizability for integer sequences, applying them to classical sequences like Euler and Bernoulli numbers to uncover their structural and dynamical properties.

Contribution

It defines realizability concepts for sequences and applies them to classical sequences, revealing new dynamical and algebraic insights, including characterizations of Bernoulli primes.

Findings

Bernoulli denominators are algebraically realizable at every prime

Euler numbers cannot be realized on a nilpotent group

Diverse congruences are explained through algebraic realizability

Abstract

We introduce the notions of local realizability at a prime and algebraic realizability of an integer sequence. After discussing this notion in general we consider it for the Euler numbers, the Bernoulli denominators, and the Bernoulli numerators. This gives, for example, a dynamical characterization of the Bernoulli regular primes. Algebraic realizability of the Bernoulli denominators is shown at every prime, giving a different perspective on the great diversity of congruences satisfied by this sequence. We show that the sequence of Euler numbers cannot be realized on a nilpotent group, which may explain why it is less hospitable to congruence hunting.