Talk:Pisot-Vijayaraghavan number
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An anonymous contributor put this article in Category:algebraic integers which does not exist. I wonder, should one create such a category, or just put this article in Category:Algebraic number theory. Thanks. Oleg Alexandrov 03:53, 7 Mar 2005 (UTC)
Algebraic number theory is not the same as the theory of algebraic numbers; it is really the theory of algebraic number fields. I suggest a Category:Algebraic numbers, since, for example, there are results of diophantine approximation that talk about individual numbers. Charles Matthews 11:51, 7 Mar 2005 (UTC)
What's the command for "infinity" in TeX ? I spelled out "infinity" in letters in the formula lim_n x^n - a_n in the article, someone might want to fix that. --FvdP 22:49, 23 December 2005 (UTC)
[edit] Pisot numbers
The tendency among people in the field seems to be towards calling these simply Pisot numbers; hence I mention this name for them in the first sentence. Gene Ward Smith 21:29, 11 April 2006 (UTC)
[edit] Does "x^n tends to integers" imply "x is Pisot" ?
I'm a bit puzzled over this statement "The converse holds: if x is a real number > 1 and there is a sequence of integers an so that 
, then x is a Pisot-Vijayaraghavan number." It seems to be an open question not so long ago, so it may be still unknown. If it has been recently resolved, it would be nice to include a reference.
(written by User:Fedja who forgot to sign)
- You may be right. AFAIR I am the one who added that "the converse holds". I was sure I read this result somewhere, about 10 years ago or more, as proven. Not so sure now. Since I'm no specialist I may have been wrong all along. The close but more general "Pisot-Vijayaraghavan conjecture" looks still open indeed. --FvdP 20:04, 20 July 2006 (UTC)
- OK, as I could find no confirmation of it (though no clear refutation either), I removed the dubious statement. --FvdP 21:20, 17 August 2006 (UTC)
[edit] Questions
Let S denote the set of PV numbers. I have thought of a number of questions about the structure of S. Anybody know what has been published about them?
1) Is S a well-ordered set? No.
2) Let B denote the set of PV numbers that are less than the golden ratio. Does B contain just finitely many numbers of any given degree? Is the golden ratio the smallest quadratic number in P? Does B contain just 2 cubic and just 3 quintic numbers? Yes to all, answered in Bertin et al.
3) I find that the first 9 numbers of B have discriminants -23, -283, 1609, -31, 37253, 3857, -691055, 29077,and 4477. I wonder whether every number in B has a symmetric group as its Galois group. The first 9 numbers do, usually because the discriminant has an unsquared prime factor.
4) Does B contain any non-units? No.
5) What kind of numbers are limit numbers in S?
Scott Tillinghast, Houston TX 02:18, 7 March 2007 (UTC) Revised Scott Tillinghast, Houston TX 07:33, 10 March 2007 (UTC) Revised Scott Tillinghast, Houston TX 08:52, 8 April 2007 (UTC)
