#### Document Type

Dissertation

#### Date of Degree

Summer 2013

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Mathematics

#### First Advisor

Daniel D. Anderson

#### Abstract

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.*a*_{1}.*a*_{2}...*a*_{t} where *t* ≥ 1, Λ= 1 or -1 and the nonunit nonzero integers *a*_{1},*a*_{2},...,*a*_{t} satisfy *a*_{1} ≡ *a*_{2} ≡ ... ≡ *a*_{t} mod *n*. The *Τ*_{n}-factorizations of the form *a* = *a*_{1},*a*_{2},...,*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.

#### Pages

v, 54 pages

#### Bibliography

Includes bibliographical references (page 54).

#### Copyright

Copyright 2013 Alina Florescu