On the Characterization of ${\tau }_{(n)}$-Atoms

In 2011, Anderson and Frazier define the concept of ${\tau }_{(n)}$-factorization, where ${\tau }_{(n)}$ is a restriction of the modulo $n$ equivalence relation. These relations have been worked mostly for small values of $n$. However, it is sometimes difficult to extend findings to larger values of $n$. One of these problems is finding ${\tau }_{(n)}$-irreducible elements or ${\tau }_{(n)}$-atoms in order to characterize elements that have a ${\tau }_{(n)}$-factorization in ${\tau }_{(n)}$-atoms. The ${\tau }_{(n)}$-irreducible elements are well known for $n=0,1,2,3,4,5,6,8,10,12$. However, the problem of determining the ${\tau }_{(n)}$-atoms becomes much more difficult the larger $n$ is. In this work, we present an algorithm to construct families of ${\tau }_{(n)}$-atoms. It is shown that the algorithm terminates in finitely many steps when $n$ is the safe prime associated to a Sophie Germain prime.