Number theory is the branch of mathematics that studies properties of the integers. Despite being around for thousands of years, there are still many fundamental unsolved problems. In this class we will cover some of the basic techniques of number theory, including modular arithmetic, Fermat’s Little Theorem, primes and unique factorization. These can help you understand where divisibility rules come from, how primality tests work and how to solve systems of modular equations.

You've seen the complex numbers: they form a system of arithmetic where each number is specified by a pair of real numbers. Can you figure out a way to add and multiply triples of real numbers? Quadruples? It turns out that if you want all of the rules of arithmetic to hold, you're out of luck. But if you're willing to relax the requirement that $xy=yx$, then it's possible to define a way of multiplying quadruples of real numbers (but not triples!) in a way that gets you everything else you'd want. We'll construct these quaternions, discuss why it doesn't work in any dimension other than 4, and apply them to problems like efficiently rotating coordinates in 3 dimensions and generalize them to algebras over other fields. This will be a difficult class: come prepared for quite a bit of abstraction.

$$719 = 26^2 + 5^2 + 3^2 + 3^2$$. Can you write it as a sum of three squares? Can you write $$2007$$ as a sum of four squares? As it turns out, every positive integer can be written as the sum of four squares. We'll prove this result using a method known as infinite descent, and then proceed to discuss the number of ways a given $$n$$ can be represented so. As for a sum of three squares, we'll see which numbers can be represented in this fashion along the way. See if you can figure out the answer now (and prove it too). We'll use the tools developed in my Number Theory class. This course will not be as hard as Quaternion Algebras, but it will still be reasonably difficult, especially near the end.

Number theory is the branch of mathematics that studies properties of the integers (..., -2, -1, 0, 1, 2,...). Despite being around for thousands of years, there are still many fundamental unsolved problems. Come learn about modular arithmetic, which helps explain quick methods for testing if a number is divisible by 11. Or Fermat's Little Theorem, which forms the basis for a powerful technique for quickly checking if a number is prime.

You've learned how to take square roots: $\sqrt{4} = 2$. But what about $\sqrt{-4}$? You may have been taught that taking square roots of negative is impossible. This isn't quite true: you can enlarge your notion of what a number is in a way that allows one to compute with numbers including $\sqrt{-4}$. Such "complex numbers" are crucial all over mathematics, engineering and physics. Come and see why!

The algebra you learned in middle school is only the beginning of the story. You've learned how to manipulate symbols ...

There are many possible definitions of an elliptic curve. Perhaps the easiest to understand is the following. An elliptic curve ...

Is it possible to have a system of numbers like the real numbers or rational numbers, where we can add, ...

We will continue our exploration of algebra after dinner, proceeding to more advanced topics like field theory and Galois groups. ...

This is my annual crazy course, where there's very little chance of anyone understanding much of what I talk about. ...

Tired of Monopoly, Taboo and Trivial Pursuit? This class will introduce "European-style" board games, which range from monks in a ...

In this second hour we will continue exploring p-adics by considering their extensions. p-adic extensions are much more rich and ...

Number theory is the branch of mathematics that studies properties of the integers. Despite being around for thousands of years, ...

A real number is a number like $$ ==2.73, -47, \pi, e^{7.91}\cdots== $$ But what is a real number, actually? ...

A group is a set with one operation that is associative, has an identity and in which every element has ...

The algebra that you learn in middle school and high school is a mere shadow of the subject known to ...

We will continue the study of the objects defined in Actual Algebra 1: learning more about them and giving more ...

In the integers one has unique factorization into primes: 6=2*3. But in other types of numbers, this unique factorization breaks ...

Some of the most interesting results about topological spaces can most easily (or even only) be proved using the methods ...

Come ask me math questions! Ask anything, and I'll try to answer. Obviously, I don't know everything, but I may ...

Category Theory is a language for speaking about advanced mathematics. Often described as "abstract nonsense" or "crazy moon language," it ...

You've learned how to solve equations like ==x^2 - 7y^2 = 1==: for any y just set ==x = sqrt(1+7y^2)== ...

Building on the material in Actual Algebra 1, we will consider fields in more depth. In particular, we will study ...

Number theory is the study of properties of the integers: ..., -2, -1, 0, 1, 2... Many of the basic ...

Topology is the study of spaces that you can stretch and bend. The standard joke is that a topologist is ...

A real number is a number like ==2.73, -47, \pi, e^{7.91}\cdots== But what is a real number, actually? How does ...

Prerequisites: "Number Theory," "Introduction to Group Theory" and "Rings, Fields and Vector Spaces." Does it annoy you that you always ...

Prerequisites: "Introduction to Group Theory" and "Number Theory." Elliptic curves were crucial in the proof of Fermat's Last Theorem, provide ...

Number theory is the branch of mathematics that studies properties of the integers. Despite being around for thousands of years, ...

Have you ever wondered what those vectors you learn about in physics really are? Or whether there are other systems ...

Prerequisites: Number Theory, Algebraic Structures. Does it annoy you that you always understand what's going on in your math class? ...

Groups, Rings, Fields, Vector Spaces, Modules, Algebras and possibly even sheaves! We'll go through the definitions, some examples, and some ...

Don't you hate it when your teacher tells you that "that's beyond the scope of this class" or "we don't ...

Elliptic curves were crucial in the proof of Fermat's Last Theorem, provide both a factorization and encryption algorithm and are ...

Grassmanians pop up in many areas of mathematics, boggling your mind with their impossibility to visualize. I will tell you ...

We will cover modular arithmetic, Fermat's Little Theorem, primes, unique factorization and other material from elementary number theory. There are ...

Prerequisites: Basic Number Theory,Algebraic Structures Does it annoy you that you always understand what's going on in your math class? ...

Groups, Rings, Fields, Vector Spaces, Modules, Algebras and possibly even sheaves! We'll go through the definitions, some examples, and some ...

Intended primarily as a supplement to Algebraic Number Theory. However, you're welcome to attend even if you didn't go to ...

Don't you hate it when your teacher tells you that "that's beyond the scope of this class" or "we don't ...

We will cover modular arithmetic, Fermat's Little Theorem, primes, unique factorization and other material from basic number theory. There are ...

Prerequisites: Basic NumberTheory We'll go through some of the basic algorithms of cryptography, including the substitution cipher, the one time ...

Prerequisites: multivariable calculus What do a circle, a Mobius strip, and 5-dimensional space have in common? They all locally "look ...

Do you ever wonder what mathematicians do for a living? Do they just solve equations all day? In this class ...

Does it annoy you that you always understand what's going on in your math class? Do you wish that the ...

Intended primarily as a supplement to Algebraic Number Theory. However, you're welcome to attend even if you didn't go to ...

Don't you hate it when your teacher tells you that "that's beyond the scope of this class" or "we don't ...

We will cover modular arithmetic, Fermat's Little Theorem, primes, unique factorization and other material from basic number theory. There are ...

Do you want to do better on the AMC->10, the AMC->12, the AIME or USAMO, ARML or USAMTS? I'll have ...

We'll go through some of the basic algorithms of cryptography, including the substitution cipher, the one time pad, Diffie-Hellman key ...

The Pythagorean Theorem states that a^2 + b^2 = c^2. What solutions in integers exist? Well, 3:4:5 works, as does ...

We'll talk about some of the factoring and primality testing algorithms, including trial division, Sieve of Eratosthenes, Quadratic Sieve, Lucas ...

Do you ever wonder what mathematicians do for a living? Do they just solve equations all day? In this class ...