Check out daily activities at the course calendar.  You  will find reading assignments, homework exercises, and important due dates. This calendar may change often, so return to it regularly throughout the semester.
 

Course Content:	The properties of whole numbers and the myriad patterns they display have fascinated thinkers since before recorded history.  These results form a rich and elegant theory.  We will study the divisibility properties of integers, prime numbers and their distribution amongst the integers, factorization methods, linear and polynomial congruences, quadratic residues and the theory of quadratic reciprocity, multiplicative functions and Diophantine equations.  Along the way, we shall also consider the most important recent applications of this highly abstract theory to problems in cryptography.
Time & Place:	Lindner 101, MW 4:00 - 5:15pm
Instructor:	Daniel E. Otero
Office Hours:	Hinkle 104, MWF 3:00 - 4:00, or by appointment
Phone:	745-2012 (voicemail available)
Email:	otero@xavier.edu
Textbook:	Number Theory with computer applications, by Ramanujachary Kumanduri and Cristina Romero. Prentice Hall, 1998.
Computing:	We will make substantial use of Maple for computing purposes.  Access to Maple is available through the campus network.
Grading:	A standard scale (A = 90%, B = 80%, C = 70%, D = 60%) based on a total of 500 pts.   Homework Assignments = 200 pts Midterm Exam = 100 pts Paper = 100 pts Final Exam (Mon 13 Dec 4:00pm) = 100 pts Homework assignments will be assigned and collected regularly; check the daily calendar for details.  Both exams will take-home exams; test problems will be distributed at the end of a Thursday class and collected (in bluebooks) at the beginning of the following Tuesday period.  The Department of Mathematics & Computer Science has adopted this Statement of Grading Standards which you should review.
Absences:	Attendance and participation in class is expected. No extra credit work will be assigned. If you foresee that you will not be able to attend a given class period, you must make arrangements with me beforehand to schedule a time to make up any missing work. An email or a phone message before class time is sufficient. No arrangements will be made otherwise.
Homework:	Regular assignments will appear on the daily calendar.  This calendar may change often, so return to it regularly throughout the semester.
Papers:	Students will submit one research paper; the paper is due on Monday, November 15.  Students will be asked to choose a topic by October 27.  Papers should conform to the following guidelines, listed in order of importance, and will be evaluated against these criteria:    The paper should present a comprehensively researched discussion of your topic; if your focus is biographical, then some significant mathematical accomplishments must be thoroughly discussed [35 pts].  It should be presented in a clear and coherent writing style, using correct spelling, proper punctuation, and good grammar [25 pts].  It should contain a bibliography with at least three sources (at least two of which must be print, not electronic sources) and should include citations as either footnotes or endnotes.  Any standard bilbiographic style is acceptable [25 pts].  The body of the paper is to be 5+ pages long, typed or word-processed (in a 10 or 12 point font), double-spaced, with 1 inch margins.  In addition, include a separate, unnumbered cover page, follow the body with a bibliography page, and a final blank page for my comments [15 pts].  Here is a list of topics from which you may choose.  But do not view this as an exhaustive list; feel free to suggest topics of your own.    Trace the history of Greek number theory, from the Pythagorean school through its manifestations in Euclid's Elements and Diophantus' Arithmetic.  (See Projects #1 and #3 at the end of Chapter 2 of the textbook.) Investigate the history of the Chinese Remainder Theorem and its rediscovery in Europe. Tell the story of Fermat's Last Theorem; include the contributions of Fermat, Germain, and Wiles. Describe the 4-squares problem and its solution.  (See Project #2 at the end of Chapter 14 of the texbook.) Discuss the influence of Gauss' Disquisitiones Arithmeticae on modern number theory. Trace the history of the Prime Number Theorem (see p. 22 of the textbook).  Why was it felt to be important to develop an "elementary proof" in the 20th century? What are continued fractions?  (See Chapter 11 of the textbook.) How is number theory used to generate pseudo-random numbers?  (See §8.3 and §10.2 of the textbook.) Describe how transcendental numbers can be identified by their closeness of approximation by rational numbers.  (See §13.3 of the textbook.) How does one factor integers with elliptic curves?  (See Chapter 19 of the textbook.)

More Information of Interest:

At the University of St. Andrews in Aberdeen, Scotland, is is the MacTutor History of Mathematics archive. This site contains biographies and images of most of the important mathematicians throughout history.  Generally, these are reliable sources, but some errors still persist.  As with any other site on the Web, it should be used with caution.  (For example, look at the entry for Euclid. Who is the man in the engraving? Is it really Euclid?)

The best source of historical biographies is the Dictionary of Scientific Biography (DSB). This can be found in the reference section of MacDonald Library.

One of the finest web resources in number theory is The Prime Pages, developed by Chris Caldwell at the University of Tenessee at Martin.  He discusses virtually everything well known about primes numbers at this site.
 

The World Wide WebInteractive Mathematics Server (WIMS) includes a number of fun number theory calculators.  Of special interest are

• Factoris: integer factorization
• Primes: a search tool for primes
• Goldbach: represent an odd number as the sum of two primes
• Contfrac: express a number as a continued fraction

Carlos Rivera, a chemical engineer, has devised The Prime Puzzles & Problems Connection, a storehouse of various prime problems of interest to him; it's pretty good.

The Great Internet Mersenne Prime Search (GIMPS) is a network of people (administered by George Woltman and Scott Kurowski) who loan computing resources to the search for ever larger and larger prime numbers; the best the web has to offer in the service of mathematics!

A resource for information on Fermat's Last Theorem, maintained by the American Mathematical Society.

...and a long list of more sites provided by the University of Tenessee (Knoxville) Math Archives.  Happy browsing!