4x^2 -4xy + 7y^2 Problem

[David Wilson]

----- Original Message ----- 
From: "David Harden" <oddleehr at alum.mit.edu>
To: <seqfan at ext.jussieu.fr>
Sent: Monday, April 28, 2008 12:43 PM
Subject: 4x^2 -4xy + 7y^2 Problem


> [[Complete argument omitted for brevity]]
>
> Since there are no residue classes mod 24 produced by more than one of 
> those quadratic forms, the necessary congruential conditions for p to be 
> produced by one of those forms are also sufficient. Therefore any p == 1 
> or 7 mod 24 has p = x^2 + 6y^2 for some integers x and y. I think this is 
> what David Wilson wanted. In any case, the rest of the argument can 
> proceed using elementary modular arithmetic.
>
> ---- David
Given Dave Harden's argument that p == 1 or 7 (mod 24) <=> p = x^2 + 6y^2 
(the hard part, thank you Dave), I supplied the easy rest of the argument 
that p == 7 (mod 24) <=> p = 4x^2-4xy+7y^2 in an earlier note.

The relevant sequence can be changed to "Primes == 7 (mod 24)" or "Primes of 
form 24k+7".

On an unrelated note, interesting sequences that may not be in the OEIS are 
"differences of cubes" and "differences of positive cubes".