507 – Advanced Number Theory (Syllabus)

Mathematics 507: Advanced number theory.

Section 1.
Instructor: Andrés E. Caicedo.
Contact Information: See here.
Time: MWF 12:40-1:30 pm.
Place: M/G 124.
Office Hours: MF 11:40 am-12:30 pm 10:40-11:30 am (or by appointment).

The (admittedly impossible) goal is to discuss the following topics:

1. Divisibility
2. Congruences
3. Primitive roots
4. Quadratic reciprocity
5. Prime numbers
6. Waring’s problem
7. Geometry of numbers
8. Transcendental number theory
9. Additive combinatorics
10. Integer partitions

Recommended background: Mathematics 306: Number theory, or equivalent. It is highly desirable that you have taken prior courses in analysis and abstract algebra.

Textbook: Elementary methods in number theory. By Melvin Nathanson. Springer-Verlag (2000), ISBN: 0-387-98911-9.

At the end, this was a matter of personal taste, as my list of topics leans more towards the “analytic” than the “algebraic” side of things. This list is somewhat nonstandard. Here are some additional suggested references, both for the course and for number theory in general. I will add suggestions through the course, depending on the topic being covered:

1. A classical introduction to number theory. By Kenneth Ireland and Michael Rosen. Springer-Verlag (1990), ISBN: 0-387-97329-X. Highly recommended, this could have been our textbook.
2. Algebraic number theory and Fermat’s last theorem. By Ian Stewart and David Tall. A K Peters (2002), ISBN: 978-1568811192. This would be a nice textbook for a first course in algebraic number theory, but it requires background in Galois theory.
3. Multiplicative number theory. By Harold Davenport, revised by Hugh Montgomery. Springer-Verlag (2000), ISBN: 0-387-95097-4. This is a very nice book, but it definitely requires background in complex analysis.
4. Making transcendence transparent: an intuitive approach to classical transcendental number theory. By Edward Burger and Robert Tubbs. Springer-Verlag (2004), ISBN: 978-0387214443.
5. Additive number theory: The classical bases. By Melvyn Nathanson. Springer-Verlag (1996), ISBN: 978-0387946566.
6. Additive number theory: Inverse problems and the geometry of sumsets. By Melvyn Nathanson. Springer-Verlag (1996), ISBN: 0-387-94655-1.
7. Integer partitions. By George Andrews and Kimmo Eriksson. Cambridge University Press (2004), ISBN: 0-521-60090-1. Andrews also authored a more advanced textbook on partition theory, that requires complex analysis.

Grading: Based on homework.

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187 – Final exam

Here it is.

187 – On Ramsey theory

Given a natural number , write for the complete graph on vertices, and for the edgeless graph on vertices.

As explained in problem 47.10 from the textbook, given natural numbers , the notation

means that is so large that whenever is a graph on vertices, either contains a copy of as a subgraph, or a copy of as an induced subgraph.

We denote the negation of this statement by . In detail, this means that there is a graph on vertices such that for any collection of vertices of , at least one of the edges between them is not in , and also for any collection of vertices of , at least of the edges between them is in .

Note that if then also:

• ,
• for any , and
• for any .

Clearly, and for any . It is also clear that for any . When , however, the determination of the smallest such that is a subtle and difficult problem. This is called the Ramsey number of . We will denote it , so

and, if , then .

403/503 – Lagrange’s four squares theorem

It is a theorem of Lagrange that every natural number is the sum of 4 squares. (Since 7 is not a sum of 3 squares, 4 is best possible. Gauss showed that a number is a sum of 3 squares iff it is not of the form )

Although there are elementary proofs of this result (elementary here means in the sense of number theory. It is not a synonym for easy), I have always found the proof sketched in problem 1 of this pdf (from a course I taught a few years ago) to be quite charming. It uses a bit of the ideas we have recently discussed, so I figured you may be interested in taking a look at it.