 Hi there, it is a pleasure to welcome you to this basic course on number theory. Number theory is that branch of mathematics which deals with properties of integers. So, we all know the positive integers, these are the numbers 1, 2, 3, 4, 5 and so on. So, these are the numbers you could count if you had enough fingers at hand and then there are the negatives of these integers the minus 1, minus 2, minus 3, minus 4, minus 5 and dot, dot, dot and we have the number 0, we have the integer 0. So, all these together constitute the set of integers for us. Now we all also know that the set of integers has two basic structures defined on that which which are these structure of addition and the structure of multiplication. So, with respect to addition the set is clearly closed which means that if you take any two integers then their addition is again an integer. If you were to take 5 and minus 3 you add them you are going to get 2 which is clearly an integer. If you were to take 7 and 5 you are going to get 12 which is also an integer. So, the set of integers is closed under taking addition there is the number 0 which works like which is actually the identity as far as addition is concerned that means if you take any integer n and add 0 to it you are going to get your n back nothing is going to happen to it. Further you have the negative of any integer that means if you take any n there is the minus n with the property that the sum of these two is going to be 0 there is the associativity property. So, all these properties are put in one box which is labeled group structure. So, the set of integers is a group with respect to addition there is also the operation of multiplication which is very interesting because while the set of integers is closed under taking multiplication if you multiply any two integers you are going to get integer whether it is 0 or non-zero it does not matter you may take 2020 and multiply by 0 you are going to get 0 but 0 is an integer after all. You have the identity the one which when multiplied to any n will give you that n back and of course there is associativity if you have three integers you multiply them in any order you multiply a to b and then to this product you multiply by c or you keep a as it is first obtain bc and then obtain a into bc you are going to get the same result. But things get interesting when we come to the question of division. Now to an integer if you wanted to divide by any integer then it is not possible while remaining in the set of integers. I know some of you may be thinking that you cannot divide by 0 while that is true but you know even if I take a non-zero integer suppose I wanted to divide 2 by 5 I will get some number which is which will be called 2 by 5 somebody may call it 4 by 10 there is some number but it is not an integer that is what I mean by saying that the set of integers is not closed under taking divisions. So you cannot divide a given integer by any other non-zero integer. However the division is sometimes possible you can divide 10 by 2 you can divide 20 20 by 20 you can divide 20 20 by 101. So sometimes division is possible and this throws up a very interesting concept called the concept of a prime integer. Let me just give you the definition a prime integer or a prime number for us is that positive integer which has exactly two positive devices. So 1 is not a prime by our definition and it is not considered to be prime. One has only one divisor namely one no other positive integer is going to divide one and when we say division we want to remain in the set of positive integers. So 4 is not a prime because 1 divides 4, 2 divides 4 and 4 divides 4. So it has 3 devices for a prime we need only 2 devices. So 2 is a prime, 3 is a prime, 5 is a prime, 7 is a prime, 11, 13, 17, 19, 23, 29, 31, 37 there are lots of prime numbers. Do the prime numbers go all the way up to infinity are there infinitely many primes? Are there infinitely many primes of the form 4n plus 1? 4n plus 1 means these are those numbers which when divided by 4 leave the remainder 1. Are there infinitely many primes of the form 6n plus 5? These questions and many other interesting questions are going to be studied in this course. While we will prove that there are infinitely many prime numbers we are we will give some sort of an idea for 4n plus 1, 4n plus 3 we will not be able to do the general case where which is called the Dirichlet's prime Dirichlet's theorem on infinitude of primes in arithmetic preparations. But we will study some of these very basic properties we will also try to determine the set of primes which are sums of 2 squares the set of primes p which can be written as m square plus n square. For instance 5 is a prime which is 1 square plus 2 square, 17 is a prime which is 1 square plus 4 square, 37 can be written as 1 square plus 36 6 square 36, 13 can be written as 9 plus 4 3 square plus 2 square whereas 3 is not a sum of 2 squares, 7 is not a sum of 2 squares, 11 is not a sum of 2 squares and so on 19 is not a sum of 2 square. We will explicit 2 is a sum of 2 squares 1 square plus 1 square. So, we will explicitly determine the set of primes which are sums of 2 squares and of course the remaining ones will be the elements which cannot be written as sums of 2 squares. So, these and many other basic properties of integers mainly the properties of prime numbers will be studied in this course. What is the objective basic there? So, by basic I mean that we are going to use basic mathematics to study these. There is of course the branch of number theory called algebraic number theory where we use field theory, Galois theory and so on to understand number theory. There are the analytic number theory things which are the ones where we use complex analysis, contour integration and so on these things to study number theory. But here we are going to use basic things like counting, divisibility and some very basic group theory techniques will be used. Other than that advanced mathematical tool will not be used. However, let me warn you that this is not going to be an easy course. It is going to be an interesting course that I can promise but it is not going to be easy. You will have to work very hard to understand and learn this course. I will try to help in every possible way. My name is Sripad Gargay. I teach mathematics at IIT Bombay. I am going to run this course with the help of my teaching assistants. I hope you will join this course, enroll for this course and let us have a fun ride together. Thank you very much.