A note on class-number one in pure cubic fields

Abstract. We examine a subset of the pure cubic fields wherein individual fieldsappear to have a probability of having class-number one approximately equal to3/5. We also suggest more elaborate but more efficient algorithms that could be used to extend the data. 1. Introduction. It is known [1, p. 313] that if n distinct primes p = 1 (mod 3)divide N , the class-number h of Q(\/N) is divisible by 3". Since almost all N, i.e., allexcept for a set of measure zero, satisfy this condition, one weak consequence is thatthe density of Q(yjN) with h = 1 must equal zero. But if N is a prime q = 2 (mod 3),then 3 does not divide h, and if we restrict N to these prime radicands q it is plausible[2] that the Q(\Jq) with h = 1 now have a positive asymptotic density in thissmaller set of fields.Of the 617 primes q 104 in order to examine theconstancy of this mean density.This was done in [3]. For the 1880g < 35,000 the class-numbers of Q(\Jq) aredistributed as follows: