Material for this course will be posted on Sakai.
This course is an introduction to some of the main questions and
ideas of classical number theory, with a focus on individual
exploration and personal discovery. We'll begin by investigating the
question "What is a number?", and see how this question led to some
brilliant and celebrated breakthroughs by mathematicians such as Euler,
Fermat, and Gauss. This will culminate in an examination of modern-day
cryptography: how number theory allows transactions over the internet,
and many more things, to be made secretly and securely.
You are not assumed to have any technical mathematical knowledge beyond the basic operations of arithmetic, only a desire to understand for yourself the surprisingly mysterious properties of the natural numbers (1, 2, 3, ...). Some previous familiarity with making precise mathematical statements and working with proofs will be very helpful, but is not absolutely necessary.
The course is roughly divided into two parts. The in-class "seminar" part is a series of lectures where we explore properties of the natural numbers. This will cover the foundations of number theory as well as modern applications to cryptography. The other part is your personal exploration of some topic in number theory that branches off from the lectures, and culminates in a mathematical paper where you explain the topic in detail, as well as a 30-minute presentation to your colleagues. You'll have ample help along the way.