Number Theory

Number Theory (Fall 2018) is the study of the positive integers and closely related numbers. It was one of the first branches of mathematics to be carefully studied, with major results going back to ancient Greece and perhaps even earlier than that. Evidently, the positive integers and their mysterious properties have fascinated people for millennia, and they continue to do so today.

One of the key results in modern elementary number theory is the quadratic reciprocity theorem. Given an integer a and a prime p, we can ask whether there exists an integer x such that x^2\equiv a\pmod{p}. When this happens, we say that a is a quadratic residue modulo p, and if not then we say that a is a quadratic nonresidue modulo p. Given two primes p and q, it turns out that there is a very surprising relation between whether p is a quadratic residue modulo q and whether q is a quadratic residue modulo p. In this class, we will learn at least one proof of this famous theorem, as well as various consequences of it.

Another major question in modern number theory is the question of representability of integers by quadratic forms. Given a function f(x,y)=ax^2+bxy+cy^2 for some fixed integers a,b,c, we say that an integer n is represented by f if there are integers x,y such that f(x,y)=n. The question of representability by quadratic forms encompasses several classical questions in number theory, such as the question of which integers can be written as a sum of two squares. While the full answer to the question of which integers are represented by which quadratic forms is still not fully solved, we will look at much of what is currently known, and we will see some of the deep connections this question has to abstract algebra.

We will also look at elliptic curves and their use in number theory. An elliptic curve is a curve of the form y^2=x^3+ax+b. Many problems in number theory can be converted into problems about elliptic curves in surprising ways. One of the most famous of these is the congruent number problem: for which integers n does there exist a right triangle with rational sides and area n? It is not always easy to find these triangles when they exist; for instance, when n=5, we have to come up with the triangle with legs \frac{20}{3} and \frac{3}{2}, and hypotenuse \frac{41}{6}. We will see how to turn this problem into a question about rational points on elliptic curves, and why that is a good idea.