Vol. 32, No. 1, 1970

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On groups of linear recurrences. II. Elements of finite order

R. Robert Laxton

Vol. 32 (1970), No. 1, 173–179

For each quadratic polynomial f(x) Z[x], whose ratio of roots is not ±1, a group G(f) of equivalence classes of certain linear recurrences with companion polynomial f(x) has been constructed by the author. Its structure was shown to be connected with the structure of the sets of prime divisors of the linear recurrences. The group G(f) is infinite but its torsion subgroup is finite and usually, but not always, consists of just two elements; the class of the Lucas sequence = [0,1] of f(x) and the class of the recurrence = [2,P] associated with f(x). This subgroup is completely determined here for each polynomial f(x). In 1961 M. Ward raised the question whether () and () are the only classes whose sets of prime divisors can be characterized globally. It is shown in this article that there are groups G(f) with elements of finite order, other than () and (), whose prime divisors can be similarly characterized.

Mathematical Subject Classification
Primary: 10.61
Secondary: 20.00
Received: 11 December 1968
Revised: 22 July 1969
Published: 1 January 1970
R. Robert Laxton