Talk:List of number fields with class number one

Real quadratic fields of prime discriminant

I'm a number theoretic ignoramus, but it struck me that the list of real quadratic fields in this article contains all prime numbers that are 1 modulo 4 in the range displayed (namely 5,13,17,29,37,41,53,61,73,89,97). This struck me as quite odd, so I looked at the corresponding OEIS sequence, only to find that the regularity continues even to the end of that list (namely it contains 101,109,113,137,149,157,173,181, after which the sequence ends). However the OEIS does give access to a text file that goes on up to 5000, and of the couple of following Pythagorean primes (sequence 002144 in the OEIS) 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, the numbers 229 and 257 are missing from the table, and surely there are more to follow. So this confirms what I guessed, but I would have liked the evidence to have been more authoritative. Browsing a bit in number theory books I found a statement that seems to directly relate to my observation: in "H. Cohen and H. W. Lenstra, Heuristics on class groups of number fields, Number Theory, Noordwijkerhout 1983, Proc. 13th Journées Arithmétiques, ed. H. Jager, Lect. Notes in Math. 1068, Springer-Verlag, 1984, pp. 33—62; MR0756082", I found on the first page "It seems that a definite non zero proportion of real quadratic fields of prime discriminant (close to 76%) has class number 1, although it is not even known if there are infinitely many." However this "it seems" is not supported in that paper by concrete tables, and in particular they do not make the observation that the first 21 real quadratic fields of prime discriminant all have class number one (maybe this is so well known to number theorists that it is not even worth noting?). The reason I'm asking here is that I would like to add something like "despite appearance to the contrary, this list does not contain all prime numbers that are 1 modulo 4" to this article, but I would prefer having a good reference. I know "Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, 1966" is supposed to contain a sufficiently long list, but I don't have a copy of that at hand. Marc van Leeuwen (talk) 13:58, 25 March 2011 (UTC)

Appendix B of Cohen's book A Course in Computational Algebraic Number Theory has tables of low degree number fields and class number data. In particular Appendix B.2 is a table of quadratic fields of discriminant up to 497. And indeed, it states that discriminant 229 has class number 3. You can use this as a source. Much more extensive tables are available at [1] and these are rather authoritative (they are produced by the research group that Cohen is a part of with software that Cohen helped write). As for more detailed discussion, I can get back to you on that a bit later. RobHar (talk) 14:34, 25 March 2011 (UTC)