Cycles of Sums of Integers

We study the period of the linear map $T:$\mathbb{Z}$m^n\rightarrow $\mathbb{Z}$m^n:(a_0,\dots,a{n-1})\mapsto(a_0+a_1,\dots,a{n-1}+a_0)$ as a function of $m$ and $n$, where ${\mathbb{Z}}_m$ stands for the ring of integers modulo $m$. Since this map is a variant of the Ducci sequence, several known results are adapted in the context of $T$. The main theorem of this paper states that the period modulo $m$ can be deduced from the prime factorization of $m$ and the periods of its prime factors. We also characterize the tuples that belong to a cycle when $m$ is prime.