The Fibonacci numbers are a sequence where each consecutive term is the sum of the previous two, and the first two members are 1,1. It goes 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. What would the tenth member of the sequence be if instead of 1,1 we started with any two numbers a,b and used the same rule to generate each term from the previous two? We'll learn how to use this and similar tricks (which may or may not involve matrices) to get lots of Fibonacci numbers very quickly and impress your friends.

What are the integer solutions to the equation $$61 a^2 + 1 = b^2$$? In 1657, Fermat boasted that mathematics in France is better than in Britain, and challenged British mathematicians to solve this equation (they did). In this class we'll take him up on the challenge, and show America is no worse than the Brits. I'll talk the students through solving this and other Diophantine equations, and introduce some cool relationships between number theory and geometry.

We'll learn how to take sums of certain series, including the famous identity $$\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$ We'll talk about Zeta functions (defined by series like the one above) and their relation to primes, and learn about a couple of open problems, including one of the most important and elusive conjectures in mathematics, the Riemann Hypothesis.