Ducci sequences and cyclotomic fields

作者：Breuer Florian^*

来源：Journal of Difference Equations and Applications, 2010, 16(7): 847-862.

DOI：10.1080/10236190802566509

• Ducci sequences
• cyclotomic fields
• cyclotomic polynomials
• finite abelian groups
• finite fields
• multiplicative orders
• Wieferich primes

Let G be an abelian group and n is an element of N. A Ducci sequence over G is a sequence of n-tuples u; D(u); D(2)(u), ... is an element of G(n), where D(u(1); u(2), ..., u(n)) := (u(1) + u(2); u(2) + u(3) , ... , u(n) + u(1)). When G is finite, this sequence is eventually periodic. In this paper, we study Ducci sequences over G Z/p(t)Z, where p inverted iota n, 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.

• 出版日期2010