Introduction

The purpose of this paper is to give an overview of the main recent advances concerning Gauss's class number one problem for real quadratic fields, to describe the connections with prime-producing polynomials, continued fraction theory and the theory of reduced ideals, and to make the comparison with the development of the solution of Gauss's class number one problem for complex quadratic fields. This includes a description of the search for a real quadratic field analogue of the well-known Rabinowitsch result for complex quadratic fields.

Furthermore, we describe a criterion for class number 2 (in terms of continued fractions and reduced ideals) for general real quadratic fields. We also provide (for a specific class of real quadratric fields called Richaud-Degert types) class number 2 criteria in terms of prime-producing quadratic polynomials. This is the real quadratic field analogue of Hendy's result [9] for complex quadratic fields. Other related results including a solution of a problem of L. Bernstein [2], [3] are delineated as well.