Some power-sequence terraces for ℤ pq with as few segments as possible

Abstract

A power-sequence terrace for Zn is a Zn terrace that can be partitioned into segments one of which contains merely the zero element of Zn whilst each other segment is either (a) a sequence of successive powers of an element of Zn, or (b) such a sequence multiplied throughout by a constant. If n=pq, where p and q are distinct odd primes, the minimum number of segments for such a terrace is 3+ξ(n), where ξ(n) is the ratio φ(n)/λ(n) of the number of units in Zn to the maximum order of a unit from Zn. For n=pq, general constructions are provided for power-sequence Zn terraces with 3+ξ(n) segments. These constructions are for ξ(n)=2, 4 and 6, and they produce terraces throughout the range n<200 except for n=119,161. © 2005 Elsevier B.V. All rights reserved.