Heuristics and related results on class groups of real quadratic fields

The fact is studied that the ideal class numbersh of types of real quadratic fields usually contain a fixed prime numberp as a factor, and the reason is found to be existing there a kind of prime ideals whosepth powers are principal. A modification of the Cohen-Lenstra Heuristics for the probability that in this situation the class numberh is actually a multiple ofp then is presented: Prob (p|h)=1-(1-p ^-1)(1-P ^-2)⋯. This idea is also extended to predict the probability that the classP represented by the above prime ideal is actually of orderp: Prob (_o(P)=p) =1/p. Both of these predictions agree fairly well with the numerical data. Project supported by the National Natural Science Foundation of China.