# Cycles of Sums of Integers

## TL;DR

This paper analyzes the cycle lengths of a linear map on modular integer vectors, revealing how these periods depend on the prime factorization of the modulus and characterizing cycles for prime moduli.

## Contribution

It provides a formula for the period of the map based on prime factorization and characterizes cycles when the modulus is prime.

## Key findings

- Period depends on prime factorization of m
- Cycle characterization for prime m
- Main theorem relates periods to prime factors

## Abstract

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.