Suites r\'ecurrentes lin\'eaires d'ordre 2 \`a divisibilit\'e forte

TL;DR

This paper provides a simpler, elementary proof of a 1985 result characterizing certain second-order linear recurrence sequences of integers that satisfy a strong divisibility condition, expanding understanding of their properties.

Contribution

It offers a new, simplified proof of a known classification of integer recurrence sequences with strong divisibility, originally established by Horák and Skula.

Findings

Characterization of sequences satisfying strong divisibility

Simplified proof of the classification result

Reaffirmation of properties of second-order linear recurrences

Abstract

We reprove twice, in a simpler but as elementary way, a result by Hor\'ak and Skula (1985) who determined, among all sequences of integers defined by $$u_1=1,\quad u_2=R,\quad u_{n+2}=Pu_{n+1}-Qu_n$$ for some integers $P,Q,R$, those which satisfy the strong divisibility condition $$\forall i,j\in\mathbb N^*\quad u_i\land u_j=\left|u_{i\land j}\right|,$$ where $\land$ denotes the greatest common divisor.