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 Τn-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 Τn-factorization of a nonzero nonunit integer a is a factorization of the form a = Λ.a1.a2...at where t ≥ 1, Λ= 1 or -1 and the nonunit nonzero integers a1,a2,...,at satisfy a1a2 ≡ ... ≡ at mod n. The Τn-factorizations of the form a = a1,a2,...,at (that is, without a leading -1) are called reduced Τn-factorizations. While similarities exist between the Τn-factorizations and the reduced Τn-factorizations, the study of one type of factorization does not elucidate the other. This work serves to compare the Τn-factorizations of the integers with the reduced Τn-factorizations in Z and the Τn-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 Τn-factorizations in Z have been discussed by S. Hamon. Using different methods based on group theory we explore similar results about the reduced Τn-factorizations in Z and the Τn-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

Included in

Mathematics Commons

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