**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)

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.

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

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 . . .").

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).

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 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).

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.

- 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).

- 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.

Last update: November 26, 2019