Reduced tau(n) factorizations in Z and tau(n)-factorizations in N

In this dissertation we expand on the study of taun-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 taun-factorization of a nonzero nonunit integer a is a factorization of the form a = lambda· a1·a2··· at where t ≥ 1, lambda = 1 or -1 and the nonunit nonzero integers a1, a2, ..., at satisfy a1 ≡ a2 ≡ ... ≡ at mod n. The taun-factorizations of the form a = a1· a2···at (that is, without a leading -1) are called reduced tau n-factorizations. While similarities exist between the tau n-factorizations and the reduced taun-factorizations, the study of one type of factorization does not elucidate the other. This work serves to compare the taun-factorizations of the integers with the reduced taun-factorizations in Z and the taun-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 tau n-factorizations in Z have been discussed by S. Hamon. Using different methods based on group theory we explore similar results about the reduced tau n-factorizations in Z and the taun-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.;Using our approach the taun-factorizations in N are shown to be the easiest to describe. In Z the taun-factorizations pose less of a challenge than the reduced taun-factorizations.