A Study of the Anatomy of the Integers via Large Prime Factors and an Application to Numerical Factorization

Permanent Link:

http://ufdc.ufl.edu/UFE0045058/00001

Material Information

Title:

A Study of the Anatomy of the Integers via Large Prime Factors and an Application to Numerical Factorization

Physical Description:

1 online resource (111 p.)

Language:

english

Creator:

Molnar, Todd W

Publisher:

University of Florida

Place of Publication:

Gainesville, Fla.

Publication Date:

2012

Thesis/Dissertation Information

Degree:

Master's ( M.S.)

Degree Grantor:

University of Florida

Degree Disciplines:

Mathematics

Committee Chair:

Alladi, Krishnaswa

Committee Members:

Rosalsky, Andrew J
Sin, Peter

Subjects

Subjects / Keywords:

anatomy -- factorization -- factors -- integers -- large -- prime
Mathematics -- Dissertations, Academic -- UF

Genre:

Mathematics thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:

Ostensibly, this thesis attempts to discuss some important aspects of the theory surrounding the anatomy of the integers (to borrow a term from de Koninck, Granville, and Luca), by which I mean the theory surrounding the prime factorization of an integer, and the properties of these prime factors. The anatomy of the integers is a very dense topic which has attracted the attention of theoretical and applied mathematicians for many years, due largely to the fact that the questions involved are often difficult and can be approached from many different angles. Virtually every branch of mathematics has benefited from the study of the prime factorization of the integers including (but certainly not limited to) number theory, combinatorics, algebra, and ergodic theory (to name but a few). This thesis will concern itself with both the theoretical and computational aspects of this study, and will use that theory to understand the algorithmic factorization technique introduced by Knuth and Pardo in 1976. To fully understand this algorithmic procedure, we will discuss a good deal of theory related to the distribution of prime numbers and the largest prime factor of an integer. The thesis is divided into four chapters, Chapter 1 is an introduction which gives a brief overview of the motivation and rich history surrounding these deep problems. Chapter 2 sketches two different proofs of the Prime Number Theorem, which is an indispensable tool in addressing problems related to integer factorization, as well as containing several comments related to (currently) unproven results concerning the distribution of primes (in particular, the Riemann hypothesis). One of the proofs of the Prime Number Theorem we have included in this section appears novel, although it undoubtedly could be deduced by any individual with sufficient knowledge of analytic number theory. Following the work of Alladi, Erdos, Knuth, and Pardo, Chapter 3 develops the necessary theoretical results concerning the largest prime factors of an integer. To this end, proofs of the average value of the Alladi-Erdos functions, the average value of the largest prime factor, a proof of a special case of Alladi’s duality principle, and certain density estimates are supplied. The proofs presented here related to the Alladi-Erdos functions differ from their original proofs in that we use the theory of complex variables, whereas in their original paper Alladi and Erdos derive their results elementarily. This allows us to improve their estimates for bounded functions, as well as showing the connection of these estimates with the Riemann hypothesis. The proof of Alladi’s duality principle is also derived in an analytic fashion, although not in its most general form. This differs from Alladi’s original treatment of the problem which is entirely elementary; however, in using analytic techniques we need to impose certain bounds on the arithmetic functions in question, whereas the elementary approach holds unconditionally. Further results on the largest prime factors of integers are also included, but they are due to Knuth and Pardo. The final section, Chapter 4, formally introduces the Knuth-Pardo factorization algorithm and includes their proof that the probability of a random integer between 1 and N with k -th largest prime factor < N^x , for any x greater than 0, approaches a limiting distribution; furthermore, we quote several results from the paper of Knuth and Pardo that will prove useful for studying their algorithm.

General Note:

In the series University of Florida Digital Collections.

General Note:

Includes vita.

Bibliography:

Includes bibliographical references.

Source of Description:

Description based on online resource; title from PDF title page.

Source of Description:

This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.

Statement of Responsibility:

by Todd W Molnar.

Thesis:

Thesis (M.S.)--University of Florida, 2012.

Local:

Adviser: Alladi, Krishnaswa.

Electronic Access:

RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-12-31

Record Information

Source Institution:

UFRGP

Rights Management:

Applicable rights reserved.

Classification:

lcc - LD1780 2012

System ID:

UFE0045058:00001