Date of Degree
PhD (Doctor of Philosophy)
Daniel D. Anderson
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 a1 ≡ a2 ≡ ... ≡ 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.
v, 54 pages
Includes bibliographical references (page 54).
Copyright 2013 Alina Florescu