On Gauss’s class number problems

On Gauss's Class Number Problems

Let h be the class number of binary quadratic forms (in Gauss's formulation). All negative determinants having some h = On ± 1 can be determined constructively: for h = 5 there are four such determinants; for h = 7, six; for A = 11, four; and for h = 13, six. The distinction between class numbers for determinants and for discriminants is discussed and some data are given. The question of one cl...

Class-number Problems for Cubic Number Fields

The Gauss Class-Number Problems

In Articles 303 and 304 of his 1801 Disquisitiones Arithmeticae [Gau86], Gauss put forward several conjectures that continue to occupy us to this day. Gauss stated his conjectures in the language of binary quadratic forms (of even discriminant only, a complication that was later dispensed with). Since Dedekind’s time, these conjectures have been phrased in the language of quadratic fields. This...

On a Class Number Formula for Real Quadratic Number Fields

For an even Dirichlet character , we obtain a formula for L(1;) in terms of a sum of Dirichlet L-series evaluated at s = 2 and s = 3 and a rapidly convergent numerical series involving the central binomial coeecients. We then derive a class number formula for real quadratic number elds by taking L(s;) to be the quadratic L-series associated with these elds.

Problems and Results on Combinatorial Number Theory

I will discuss in this paper number theoretic problems which are of combinatorial nature. I certainly do not claim to cover the field completely and the paper will be biased heavily towards problems considered by me and my collaborators . Combinatorial methods have often been used successfully in number theory (e.g . sieve methods), but here we will try to restrict ourselves to problems which t...