Number Theory – Introduction (1/22)

1. Number Theory – Introduction (1/22) • Very general question: What is mathematics? • Possible answer: The search for structure and patterns in the universe. • Question: What is Number Theory? • Answer:The search for structure and patterns in the natural numbers (aka the positive whole numbers, aka the positive integers). • Note: In general, in this course, when we say “number”, we mean natural number (as opposed to rational number, real number, complex number, etc.).

2. Some Sample Problems in Number Theory • Can any number be written a sum of square numbers? • Can any number be written as sum of just 2 square numbers? Experiment! See any patterns? • Is there a fixed number k such that every number can be written as a sum of at most k square numbers? • Same question as the last for cubes, quartics(i.e., 4th powers), etc. • This general problem is called the Waring Problem.

3. More problems.... • Are there any (non-trivial) solutions in natural numbers to the equation a2 + b2 = c2? If so, are there only finitely many, or are the infinitely many? • Are there any (non-trivial) solutions in natural numbers to the equation a3 + b3= c3? If so, are there only finitely many, or are the infinitely many? • For any k > 2, are there any (non-trivial) solutions in natural numbers to the equation ak+ bk= ck? If so, are there only finitely many, or are the infinitely many? • This last problem is called Fermat’s Last Theorem. • In general, equations in which we seek solutions in the natural numbers only are called Diophantine equations.

4. And yet more... Primes! • Definition. A natural number > 1 is called prime if.... • Are there infinitely many primes? • About how many primes are there below a given number n? (The answer is called the Prime Number Theorem.) • Definition. Two primes are called twins if they differ by 2.(Examples?) • Are there infinitely many twin primes? (This is, of course, called the Twin Primes Problem.) • Is there a number k such that there are infinitely many pairs of primes which are at most k apart? • The existence of such a number k was proved this past summer!!!

5. More with primes • Definition. We say two numbers a and b are congruent modulo m if m divides b – a. • Are there infinitely many primes which are congruent to 1 modulo 4? To 2 modulo 4? To 3 modulo 4? • Can every even number be written as a sum of two primes? (This is called the Goldbach Problem.) • Can every odd number be written as a sum of three primes? (This – sort of - is called Vinogradov’s Theorem.)

6. Assignment for Friday • Obtain the text. • Read the Introduction and Chapter 1. • In Chapter 1 try out Exercises 1, 2, 3 and 5.