No, this class will not be about the PRIMES program run by Pasha and Slava! My goal is to discuss prime numbers – those integers ≥ 2 that are only divisible by 1 and themselves. For example, 2, 3, 5, 7, 11, and 13 are primes. They are the building blocks of all other integers through multiplication. It's a deceivingly simple notion - but their properties are amazingly deep, some of them related to the most important unsolved problems in mathematics.
We are not going to go that far – our goal will be to learn how to use logic and, based only on the axioms of arithmetic, prove a bunch of fun facts about primes. Such as:
the fact that there are infinitely many of them among all integers, as well as among some arithmetic progressions
the Fundamental Theorem of Arithmetic (FTA): every natural number can be UNIQUELY expressed as a product of prime factors (yes, we all heard about it, but how to prove it?)
some elementary theorems in number theory that can be proved using FTA, e.g. the irrationality of the square root of 2, or the insolvability of x^3 + y^3 = z^3 in integers (a special case of Fermat's Last Theorem)
if time allows: some weak forms of Prime Number Theorem, a fundamental result in number theory that gives an asymptotic formula for the number of primes from 1 to N as N goes to infinity
of course at the end I will mention several difficult theorems about prime numbers, as well as some well-known unproved conjectures.
The class will not be just in a lecture mode, we will do a lot of stuff together, including both numerical experiments and rigorous mathematical proofs.
See you at the camp soon! Dima