Reduced τn-factorizations in z and τn-factorizations in N

In this dissertation we expand on the study of Τ_n-factorizations or generalized integer factorizations introduced by D.D. Anderson and A. Frazier and examined by S. Hamon. Fixing a non-negative integer n, a Τ_n-factorization of a nonzero nonunit integer a is a factorization of the form a = Λ.a₁.a₂...a_t where t ≥ 1, Λ= 1 or -1 and the nonunit nonzero integers a₁,a₂,...,a_t satisfy a₁ ≡ a₂ ≡ ... ≡ a_t mod n. The Τ_n-factorizations of the form a = a₁,a₂,...,a_t (that is, without a leading -1) are called reduced Τ_n-factorizations. While similarities exist between the Τ_n-factorizations and the reduced Τ_n-factorizations, the study of one type of factorization does not elucidate the other. This work serves to compare the Τ_n-factorizations of the integers with the reduced Τ_n-factorizations in Z and the Τ_n-factorizations in N.

One of the main goals is to explore how the Fundamental Theorem of Arithmetic extends to these generalized factorizations. Results regarding the Τ_n-factorizations in Z have been discussed by S. Hamon. Using different methods based on group theory we explore similar results about the reduced Τ_n-factorizations in Z and the Τ_n-factorizations in N. In other words, we identify the few values of n for which every integer can be expressed as a product of the irreducible elements related to these factorizations and indicate when one can do so uniquely.