Talk:163 (number)
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Note
[edit]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.
John McKay24.200.155.110 (talk) 15:36, 26 November 2008 (UTC)
The Slovene version of this page has some very interesting-looking observations about this number. I wish someone could translate. PrimeFan 21:44, 30 Jul 2004 (UTC)
163 and subfields of cyclotomic fields without unique factorization
[edit]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.
So, does someone know about this: is 1. true? And what ref. (best in the Web) can be given for it? --UKe-CH (talk) 23:53, 19 January 2009 (UTC)
Dead link
[edit]During several automated bot runs the following external link was found to be unavailable. Please check if the link is in fact down and fix or remove it in that case!
- http://math.arizona.edu/~mcleman/CoolNumbers/CoolNumbers.html
- In 163 (number) on 2011-05-25 01:52:13, 404 Not Found
- In 163 (number) on 2011-06-01 18:56:12, 404 Not Found