This is a bit fluffier of a question than I usually aim for, so apologies in advance if this doesn't pass the smell test for suitability.
Likely my favorite fun fact in all of number theory is the juxtaposition of two "extremal and opposite" properties about the prime 163 in relation to class numbers:
- $p=163$ is the largest value of $p$ for which the quadratic imaginary number field $\mathbb{Q}(\sqrt{-p})$ has class number equal to one. (Baker-Heegner-Stark)
- $p=163$ is the smallest value of $p$ for which the real cyclotomic field $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ has class number greater than one. (Schoof)
Of the various ways I know of "pushing up" and "pushing down" class numbers (class field theory, Herglotz-type formulas, Scholz-type reflection theorems), none seem to give any indication that these two class numbers should be related, let alone inversely so. Of course, since the smaller Heegner discriminants don't correspond to analogous real cyclotomic fields with positive class number, this is not surprising.
This leads me to wonder if there's an analytic link between these two quantities -- for example, by relating their zeta-functions and looking at the corresponding class number formulas. My initial, admittedly naive, attempts to extract anything from the relationship between the zeta-functions for $\mathbb{Q}(\zeta_p)$ and $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ have come up empty. So my question is:
Are there fancier analytic (or other) techniques that might shed some light on the "miracle" above?
Of course, I'm also aware that this juxtaposition might be purely coincidental, a mildly large example of the law of small numbers at work. I might even prefer it that way.
Edit to incorporate some computations and comments from below.
For primes congruent to 7 mod 12, the real cyclotomic field of conductor p contains a unique cyclic cubic subfield. By class field theory, there is a surjection of class groups from the real cyclotomic to the cubic. Since for 163, the cyclic cubic has class number 4, the non-triviality of the class number for the real cyclotomic can be said to "come from" the cubic. In a sense, the coincidence thus reduces to the fact that the first cyclic cubic field ("first" with respect to ordering by conductor) of prime conductor 7 mod 12 with non-trivial class group is the one of conductor 163. The fact that 163 is only the 11th prime in this congruence class may modify (in which direction I'm not sure) your opinion of whether or not this is a coincidence.
Barring any insight as to why the class number of this quadratic and cubic would be related, which may be unlikely given Franz Lemmermeyer's answer, it would be interesting to know if one could devise a clever probabilistic test for evaluating how surprised one should be to see ten class-number-one cubics in a row. I imagine that it's not very unlikely -- I just ran a computation, and class number 1 seems to very prevalent for cyclic cubics of small conductor ($p<5000$), and some heuristic (sorry, Andrew) evidence in the literature seems to agree.