From: "Everett W. Howe" <see@my.sig>
Subject: Re: A ternary quadratic form
Date: Thu, 18 Nov 1999 03:06:31 GMT
Newsgroups: sci.math.research
Keywords: representation of integers by  x^2  + 3(y^2 + z^2)

"Imre Z. Ruzsa" wrote:

> Recently I came across the following question: which positive integers
> can be represented as x^2  + 3(y^2 + z^2) , with integers x,y,z. It is
> easy to see that no integer of the form 9^k (3a+2)  has such a
> representation, and I conjecture that every other integer has one.
> Some calculations support this.

> I looked in the literature and found results on many similiar forms, but
> not on this one.  I would appreciate any reference.

This is Theorem IV (page 65) of

   L. E. Dickson, Integers represented by positive ternary quadratic
   forms, Bull. Amer. Math. Soc. 33 (1927), 63--70.

> Karoly Boroczki and I could apply this to the following question: for
> what values of k and n is there a convex lattice hexagon with k lattice
> points inside and n on the boundary.

What can you prove?

   -- Everett

________________________________________________________________________
Everett Howe                          Center for Communications Research
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domain-name: alumni.caltech.edu                     San Diego, CA  92121