Ducci sequences and cyclotomic fields
Florian Breuer · The Journal of Difference Equations and Applications · 2010
Let G be an abelian group and . A Ducci sequence over G is a sequence of n-tuples , where . When G is finite, this sequence is eventually periodic. In this paper, we study Ducci sequences over , where , using properties of cyclotomic polynomials. We first characterize the vanishing sequences (i.e. those for which the cyclic part consists only of 0), as well as the tuples in the cyclic part of a sequence. We then derive an expression for the period of a given Ducci sequence in terms of orders of roots of unity plus one modulo powers of a prime above p in a cyclotomic number field. Lastly, the dependence of the period on t leads us to a connection with Wieferich primes.