ESP Biography

ELI FONSECA, ESP Teacher

Major: Math and Physics

College/Employer: Columbia University

This course will explore which primes can be represented as $$x^2+ny^2$$ with $$x,y$$ integers and $$n$$ a natural number. Despite the questions simple appearance finding a complete answer is anything but simple as it requires a deep understanding of field class theory, modular functions, and complex multiplication. After introducing some basic preliminaries we will focus on the simpler case of $$n=1$$. I will present a constructive proof, due to Euler, which fully answers the question in this simpler case. Then we will turn our attention to the more general question by looking at a few more special cases and then present a theorem, due to Weber, that answers the general question without proof.