Group structure and maximal division for cubic recursions with a double root

Christian Jean-Claude Ballot

Vol. 173 (1996), No. 2, 337–355

DOI: 10.2140/pjm.1996.173.337

Abstract

An equivalence relation is defined on the set of recurring sequences with given cubic characteristic polynomial f(X) = (X − 𝜃₁)²(X − 𝜃₂) ∈ ℤ[X] so that sequences in a class share the same maximal prime divisors. The set G(f) of equivalence classes is shown to form a group structure by exhibiting an isomorphism φ between G(f) and the Laxton group G(f₁), where f₁(X) = (X −𝜃₁)(X −𝜃₂) is the squarefree part of f(X). The map φ has the additional property that the maximal prime divisors of a 𝒰∈ G(f) are the prime divisors of φ(𝒰) ∈ G(f₁).

Mathematical Subject Classification 2000

Primary: 11B37

Secondary: 11N64

Milestones

Received: 12 November 1993

Published: 1 April 1996