Modular periodicity of the Euler numbers and a sequence by Arnold

TL;DR

This paper investigates the periodicity of Euler numbers modulo various integers, proposing conjectures for minimal periods and initial periodic points, and introduces a sequence related to Arnold with a conjecture for its computation.

Contribution

It formulates precise conjectures on the periodicity of Euler numbers modulo q and introduces a new sequence associated with Arnold, including conjectures for their computation.

Findings

Sequences are ultimately periodic modulo q.

Conjectures on minimal period and start point of periodicity.

A new sequence related to Arnold's work with a proposed computation method.

Abstract

For any positive integer $q$, the sequence of the Euler up/down numbers reduced modulo $q$ was proved to be ultimately periodic by Knuth and Buckholtz. Based on computer simulations, we state for each value of $q$ precise conjectures for the minimal period and for the position at which the sequence starts being periodic. When $q$ is a power of $2$, a sequence defined by Arnold appears, and we formulate a conjecture for a simple computation of this sequence.