# SC 2016 Semilab: Number Theory through Fun Facts and Problems

by Pavel Etingof

Everyone knows that math studies numbers. Yet there is a separate area of math called NUMBER THEORY. What is it? Why do we need it? Why is it that the sequence of whole numbers 1, 2, 3, 4,... - seemingly one of the simplest and best studied mathematical objects - keeps challenging mathematicians with unproved conjectures, unexpected results, and easy-to-state problems which are incredibly hard to solve? For instance, many people know that any whole number can be uniquely factorized into primes. This seems obvious - just find the smallest divisor, divide your number by it, and go on until you get 1. But how to prove the factorization is unique? Well, this one is not so hard, but it is less obvious than you may think!

In this semilab we will discuss the basics of number theory - divisibility, primes, the fundamental theorem of arithmetic and its applications, modular arithmetic, Pythagorian triples, and other fun things about numbers. We will also discuss how to PROVE these things.

This semilab does not require much special background and should be accessible to all campers who are interested in math and like to solve problems, although some experience with math beyond ordinary school program is very helpful. It is based on a minicourse developed by Prof. Dmitry Kleinbock (Brandeis University).