# [seqfan] Re: A property of 163

victor miller victorsmiller at gmail.com
Fri Jul 10 16:16:49 CEST 2009

Yes, that all looks correct.  The sequence is one conjectured by
Gauss, and finally proved complete (i.e. all of the quadratic
imaginary fields of class number 1 are in that sequence) by Heegner.
His proof, however, was not believed and it was later proved
independently by two different methods by Harold Stark and Alan Baker.
There are related finite sequence of the set of quadratic imaginary
fields of class number k, for various k.  I don't remember the largest
k for which the entire sequences have been determined, but I believe
that it's now of the order of 100 (or possibly more).

Victor

On Fri, Jul 10, 2009 at 9:50 AM, Tanya
Khovanova<mathoflove-seqfan at yahoo.com> wrote:
>
> Dear SeqFans,
>
> I received the following submission for my number gossip page (numbergossip.com) from Anand Deopurkar:
>
> "A unique property of 163: It is the largest number n such that the integers in the imaginary quadratic extension Q(\sqrt -n) have the unique factorization property."
>
> Can some confirm this?
>
> He also sent a sequence which is not in the database:
>
> "Integers in the following imaginary quadratic fields Q(\sqrt -n) have the unique factorization property: n = 1,2,4,7,11,19,43,67,163. So you could add this as a rare property for those integers as well."
>
> Should we add the sequence?
>
> Tanya
>
>
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