Ducci-sequences and cyclotomic polynomials
A Ducci sequence generated by \(X=(x_1,\ldots,x_n)\in {\mathbb Z}^n\) is the sequence \((X,DX,D^2X,\ldots)\), where \(DX=(| x_1-x_2| ,| x_2-x_3| ,\ldots,| x_n-x_1| )\). Every Ducci sequence gives rise to a cycle; i.e., there are integers \(i<j\) such that \(D^iX=D^jX\). When \(i\) and \(j\) are as small as possible, the Ducci sequence has period \(j-i\). It is known that for a given cycle all entries in all \(n\)-tuples are equal to either \(0\) or to a constant \(C\). Thus, one can study such periods by working with Ducci sequences in \({\mathbb F}_2^n\). In the paper under review, the authors determine the period of a given Ducci sequence in terms of the order of a certain polynomial related to the cyclotomic polynomials, as well as to the multiplicative orders of certain elements in finite fields of characteristic \(2\). They also compute several examples and study links between Ducci sequences, primitive polynomials and Artin's primtive root conjecture. The paper concludes with several interesting open problems.
scientific article; zbMATH DE number 5347269