Lower Bounds on Class Numbers
Mentor:Jeffrey Stopple , Associate Professor of Math, University of California Santa Barbara
Which prime numbers can be written as a sum of two squares? Or as a square plus twice a square? Expressions like x^2+y^2 or x^2+2y^2 are examples of binary quadratic forms. The general case is ax^2+bxy+cy^2 with d=b^2-4ac the discriminant. The question of how many forms exist for a given discriminant goes back to Gauss. Lower bounds which are conditional on the Generalized Riemann Hypothesis were given by Hecke in 1916. In 1980, Pintz gave a nice elementary exposition of Hecke's Theorem, with explicit constants. We discuss Pintz's exposition of this theorem as well as make some slight alterations that improve the lower bound.