Dr. Tom Hagedorn will give a talk on the Primes of the Form x^2+〖ny〗^2 and the Geometry of Euler’s Convenient Numbers on November 16th from 3-4pm in room SCP 229. Refreshements will be served.
ABSTRACT: Calculation shows that 13=2^2+3^2 and 37=1^2+6^2. Which other primes p can be written as the sum of two squares? In 1640, Fermat stated that this happens precisely when p has the form 4n+1. Fermat and Euler then studied the more general question: When can a prime p be expressed in the form x^2+ny^2 ? Euler conjectured and Gauss proved that there are 65 integer values of n for which this question can be answered (I will make this precise in the talk). Such n are called convenient numbers. It is an open question if there is a 66th convenient number n, but it is known that there are at most 67 convenient numbers. This question is one of the oldest unsolved problems in number theory.
In this talk, I will give a historical overview of this conjecture and describe a new proof of Gauss’s result that uses the Geometry of Numbers. Knowledge of number theory is helpful but not required to understand the talk.