Cycles of Sums of Integers

We study the period of the linear map T:Z_m^n --> Z_m^n:(a_0,...,a_{n-1}) --> (a_0+a_1,...,a_{n-1}+a_0) as a
function of m and n, where 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.