Gauss conjectured that there are infinitely many real quadratic fields with class number one. Today this is still an open problem. Moreover, as Dorian Goldfeld, one of the recipients of the 1987 Cole prize in number theory (for his work on another problem going back to Gauss) recently stated in his acceptance of the award: “This problem appears quite intractible at the moment.” However there has recently been a search for conditions which are tantamount to class number one for real quadratic fields. This may be viewed as an effort to shift the focus of the problem in order to understand more clearly the inherent difficulties, and to reveal some other beautiful interrelationships.