It is fun to note that the square of exp(Pi.sqrt(163)) is close to an integer as are the kth powers for k up to 6. This is an example of a pseudo-obvious result.
163 and subfields of cyclotomic fields without unique factorization
The target page of the external link (about 163 as a cool number) mentions that 163 is (not only the largest Heegner number, but also) the smallest p (by which he might mean a prime) such that some kind of subfield of the p-th cyclotomic field does not have unique factorization. There seems to exist a similar result for another kind of subfield of the same field. Details:
About 15 years ago, I was looking at this unique fact. thing (in number fields). After quadratic number fields, the next simple ones seem to be normal cubic extensions of the rational field; being Galois extensions with a cyclic Galois group (of 3 elements) they are abelian, so must be subfields of a cyclotomic field. Taking for a prime p the p-th cyclotomic field, this contains an (unique) cubic field iff 3 divides phi(p) = p-1, i.e. p is of the form 6k+1. Call this field C(p); I was looking for a p=6k+1 with an 'interesting' ideal class group of C(p); call this group G(p). My first serious candidate was p=163, because I had found by individual treatment - on rather elementary grounds after deriving general formulas for C(p) - one of two things (I don't remember which): 1. for p<163, 3|p-1, G(p) is trivial, i.e. C(p) has unique fact.; 2. for these values of p, G(p) is cyclic. (Of course, 1. implies 2.) For the case p=163 I searched existing literature and so found a proof that G(163) is not trivial - from what I knew, it followed that this group is the direct product of two subgroups of order 2 (therefore not cyclic). In case 1. is true, 163 is the smallest p=6k+1 such that C(p) is without unique factorization.
My intention being to introduce such a result into one of the articles on 163 (this one here or the French one), I searched again in the math. literature to get a reference, but only in the Web using Google. Until now I only found a web page that states among other things: the class number of C(p) - which is the order of G(p) - is a multiple of 4 iff some condition is satified, and 163 is the least such p=6k+1. I know from my first lit. search on this subject that it has been studied with some intensity.
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