Number Theory

Basic Information

• Instructor: Jeff Thunder
• Office: WH 320
• e-mail: jthunder@niu.edu
• Office Hours: By appointment (though I'm often available if you simply drop by)

Course Outline and Objectives

We will be using the text An Introduction to the Theory of Numbers (5th edition) by Niven, Zuckerman and Montgomery. The material covered will be approximately the first seven chapters. The student is expected to acquire an understanding of the elementary theory of numbers. There will be some discussion of the computational aspects of these topics, but the main thrust of the course will be theoretical. You will be expected not only to follow the proofs presented in class and in the text, but also to learn to construct new proofs. Proofs must be logically correct and care must be taken to write precisely and in grammatically correct English.

Course Prerequisite

The prerequisite for this course is MATH 420. We will also use some linear algebra in places (but not much).

References

There are a plethora of books dealing with elementary number theory. Some more popular ones include (besides our text) those by Apostol, Burton, Davenport and Hardy & Wright. Check out the QA241 section in the library. Warning/inside joke: A. Weil's Basic Number Theory is not particularly well-titled ("No knowledge of number theory is presupposed in this book . . ." though "Already in Chapter 1, and throughout the book, essential use is made of the basic properties of locally compact commutative groups, including the existence and unicity of the Haar measure . . .").

Course Withdrawal

The last day for undergraduates to withdraw from the course without penalty is Friday, October 18. Graduate students can figure out the last drop day on their own (after all, you are graduate students).

Grading Scale

Grades for section 1 will be based on homework, a midterm exam and the final exam. The weights for these are 50%, 20% and 30%, respectively.

Homework

Homework will be collected once a week on Fridays. It will be turned in at the beginning of class. You are free to work with other students on the homework; in fact, this is encouraged. Sloppy and/or illegible work will be returned back with no credit! Your homework is something of which you should be proud (notice how I didn't end with a preposition there). Expect to spend lots of time on it. All of the homework problems will be checked to see that each has been done, and certain of the problems will be graded in detail, but just which problems from each assignment will be graded will not be announced in advance. The specific assignment for each week will be available on this webpage (hopefully no later than) that Monday (see below).

Exams

The midterm exam will be during class on a date yet to be determined. I try to schedule it so that you know your midterm grade before the drop deadline. The final exam is Wednesday, December 11 from 10:00 to 11:50 in the morning.

Homework Assignments

• Week #1: Numbers 1, 2, 6, 11, 14, 24, 25 and 44 from section 1.2 and numbers 10 and 31 from section 1.3 of the textbook. Print this out if you don't have access to the textbook yet.
• Week #2: Numbers 5, 6, and 7 from section 2.1 of the textbook.
• Week #3 (Due Monday, Sept. 16!): Numbers 2, 9, 13, 25, and 33 from section 2.3.
• Week #4: Number 1 from section 2.5, numbers 3 and 4 from section 2.6, and number 1 from section 2.7, and numbers 2, 3, and 9 from section 2.8.
• Week #5: Download these exercises.
• Week #6: Download these exercises.
• Week #7: Download these exercises.
• Week #8: Numbers 1, 4, 6, and 7 from section 3.2.
• Week #9: Download these exercises.
• Week #10: Number 3 from section 7.3, number 1 from section 7.4, and numbers 1 and 2 from section 7.6.
• Week #11: Download and print out this to see the homework assignment.
• Week #12: Download and print out this to see the homework assignment.
• Week #13: Good luck! No homework this week!
• Week #14: Numbers 1, 2, 4, and 8 from section 8.1 (Due Monday, Dec. 2).

Miscellaneous

• Not quite an "oldie," but certainly a "goodie:" what happens if you do really lousy on the coursework and exams?
• What's up with 8675309? We'll discuss in class.
• Some stuff about absolute values and p-adic numbers.
• Some notes on primitive roots which is a nice example of "number theory from an algebraic viewpoint."
• Here is a writeup of the material on arithmetic with polynomials that we discussed in class Sept. 20 and 23.
• Some useful applications of p-adic absolute values (the content of a polynomial, factoring polynomials with integer coefficients, and Eisenstein's Criterion).
• A very brief look at some "Higher arithmetic."
• Determining whether or not 2 is a quandratic residue.
• A proof (different than that in the textbook) for quadratic reciprocity.
• The cubic case of Fermat's "Last Theorem."
• Notes on some basic Diophantine Approximation.
• Some notes on periodic continued fractions.
• Some notes on the Pell equation.
• Some notes on basic geometry of numbers.
• Notes on Chebyshev's Inequalities.
• Notes on Bertrand's Postulate.

DRC Statement

If you need an accommodation for this class, please contact the Disability Resource Center as soon as possible. The DRC coordinates accommodations for students with disabilities. It is located on the 4th floor of the Health Services Building, and can be reached at 815-753-1303 or drc@niu.edu. Also, please contact me privately as soon as possible so we may discuss your accommodations. Please note that you will not be required to disclose your disability, only your accommodations. The sooner you let me know your needs, the sooner I can assist you in achieving your learning goals in this course.

Academic Conduct

Academic honesty and mutual respect (student with student and instructor with student) are expected in this course. Mutual respect means being on time for class and not leaving early, being prepared to give full attention to class work, not reading newspapers or other material in class, not using cell phones or pagers (pagers?!) during class time, and not looking at another student's work during exams. Academic misconduct, as defined by the Student Judicial Code, will not be treated lightly.

Homer does math!

Yes indeed, there's plenty of math humor to be found in the Simpsons. Just look and see!