Elements of Number Theory and Combinatorics

Introduction The popular view of mathematicians is that of people spending most of their life dealing with numbers. Clearly this is not all that we do although it is an important part of mathematics. These lectures will try convey to you why studying numbers is a nontrivial and often a very rewarding activity. As we growupwe learn how to count, rst up to ten and then to even larger numbers. Real life has lead us to notice certain features of numbers and has formulated us more challenging counting problems. For example, we discovered even and odd numbers. Mathematicians say that the even numbers are divisible by 2 and this is where number theory begins. Some numbers suchas 3# 5# 7# 11# 13# 17# 19# 23# 29# 31# can be quite stubborn. No matter howmuchyou try you cannot divide them into several whole equal parts larger than 1. These \stubborn" numbers are called primes and number theory tries to understand if there is a God-given pattern they satisfy. Presently, they seem to be qu