Counting in Number Theory

I.- General Information

Dates: 12 – 16 August 2019

Admission Test: An admission test was solved by the participants. You can see the test by clicking the followin button:

Note: The week will be intense since a lot of activities will be demanded from the students so they should be willing to take the activities seriously and really engage in them. It hope it will be great, but a lot of work must be put into it.

II.- Schedule.

The schedule for the first four days is as follows.

9 – 10:40CourseCourseCourseCourse
10:40 – 11BreakBreakBreakBreak
11 – 12Problems Problems Problems Problems
12 – 1Lunch Break Lunch Break Lunch Break Lunch Break
1 – 4 Problems Problems Problems Problems

The schedule for the last day is as follows:

9:10 – 10:00 Guest Lecture: Debanjana Kundu              
10:00 – 10:10 Break
10:10 – 11:00 Guest Lecture: Heejong Lee
11:00 -11:10 Break
11:10 – 12:00 Guest Lecture: Hannah Constantin
12:00 – 1:00 Lunch Break
1:00 – 3:00 5-minute talks by students
3:00 – 4:00 Tea Time

III.- Course: Counting in Number Theory

In broad terms, Number Theory is the study of numbers, mostly integers 1, 2, 3, … At first sight it is natural to wonder what we could study about them, but the truth is that there are many questions around them that are quite challenging, important and beautiful.

The basic questions of the area are like “are there an infinite number of prime numbers?”. This question was answered by Euclid and already appears in his Elements, yet many modifications of it are still studiedtoday because we have no answer for them. If we were to open a book on number theory and look at it, we would look at very strange pages full of weird claims and harsh notation. Number Theory has become a technical area, whose fame is that it is hard, because despite the easiness to ask questions in it, the tools to answer them are very hard to develop and look very different from the question they try to answer, yet if we don’t let this to daunt us we can see it is trying to answer many basic questions about the integers.

One big group of problems in the area regards the question of “counting”. What do we mean by counting? In general, we could say that counting refers to answering the following type of questions:

  1. Are there an infinite number of numbers with a certain property? For example, we know there are an infinite number of prime numbers, but we do not know there are an infinite number of Fermat primes, that is, primes of the form  . If instead we knew they are finite, then we would like to find all of them.
  2. Once we know the numbers with a certain property are infinite, we might want to understand how they are distributed. Basically, are they frequent or rare among the general integers? Even more challenging, can we find them? For example, it is a different question to ask how many integers we expect to be prime in a certain range, than giving an algorithm to find prime numbers, or even a formula that only produces prime numbers efficiently. The first question is answered by the Prime Number Theorem, while the second one is not known.

We will explore these type of questions with some examples regarding partition of integers and prime numbers, to understand why the first instinct of counting by finding the objects, in some sense that might be called combinatorial (from combinatorics, the area of mathematics that studies counting), has to be abandoned, at least partially, and move towards analytic tools where we introduce algebra and calculus to make our methods stronger, but at the expense many times of not finding explicitly the objects we are looking for.

IV.- Problem Solving and Problem Sets

Each day we will have a problem set having three parts:

  1. Part I consists of problems, some of them may be easy and others will be hard, but they are explicit questions to be thought and solved related to what we are studying in the course.
  2. Part II consists of a proof of a theorem as presented in a real book of math. The challenge will be to read it and understand it by going through the proof. This presents a very important glimpse of what the mathematician does: not only solving problems, but also reading other mathematicians write and this is challenging because there are many ways to write.
  3. Part III is an exploration. Every problem set will have one exploration, which is a big problem that is related to a direction we will not have time to study in the course but that it is interesting and that can be learned independently in the problem solving time or individually at another time.

The problem sets are not homework and are not meant to be done completely or in order, but it is expected that the students try them seriously. Talk with the other members (including the instructor) about your ideas and your progress. Write down your solutions and thoughts, it is harder than it looks to write mathematics properly.

V.- Logbook

This will be updated as the academy progresses.

DateSlidesProblem Set
August 12PDFPDF
August 13PDFPDF
August 14PDFPDF
August 15PDFPDF

VI.- Guest Lectures

We will have three guest lectures in friday given by Number Theory students in the Mathematics Department in the University of Toronto.

The abstracts for their talks are:

Guest Lectures
Debanjana Kundu.
As soon as I have them this will be updated.
Heejong Lee.
As soon as I have them this will be updated.
Hannah Constantin.
As soon as I have them this will be updated.

VII.- Five Minute Talk and student Tea Time

On Friday 16 every student will be expected to share something interested related to number theory in a five minute presentation in the board using chalk, no power point or presentations will be allowed for this. It doesn’t have to be deep or super challenging. A lot of mathematics requires presenting ideas, solutions, etc. It is part of the science and we will engage in it. Please as soon as you read this start glossing over what could you talk about, it’s free choice as long as it is inside number theory.

Possible topics could include: a proof of the infinitude of primes, talk about some open problems in number theory, famous Diophantine equations, etc. There is a world out there in number theory, so you can find something and present it!

Probably some graduate students (they are friendly) will be present for this. It is important to talk to people that know more than us and can give us feedback on what we have done, so don’t let this deter you. After we finish, we will go to the math lounge in the math department and have tea time organized by the math student union so that you have the chance to talk to students in the university about their experience!

VIII.- Conclusion

The last day we will ask every student to make a meditation on what they have learned, how they have grown during the math academy and feedback about this project and how it was handled. It will be anonymous and in questionnaire format, though we appreciate if anyone wants to tell us more!

IX.- Bibliography

You can find these books in the math library of U of T and we will have copies of most of them during the academy! You don’t need to buy anything.

BookLink in
Integer Partitions, George E. Andrews and
Kimo Eriksson
Number Theory, George E. AndrewsLink