Proof

[N. J. A. Sloane]

such assertions.
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Date: Wed, 23 Apr 2008 17:12:36 -0700
From: "Max Alekseyev" <maxale at gmail.com>
To: grafix at csl.pl
Subject: Re: Proof
Cc: "N. J. A. Sloane" <njas at research.att.com>, seqfan <seqfan at ext.jussieu.fr>
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On Wed, Apr 23, 2008 at 8:49 AM, Artur <grafix at csl.pl> wrote:
> Dear Neil,
>  I hope that I have proof:

Your proof works only in one direction. Namely, you have proved that
if a prime is of the form 4 x^2 - 4 x*y + 7 y^2 then is is also of the
form 24n + 7. To prove the converse, you need to prove the existence
of a solution in the following sub-problem:

>  2) Now if we solve equation (7 k + 7 k^2 - x - 2 k x + x^2) = m on variable
> x and we will be forced x as positive integer
>  we are receiving
>  x=(1+2k+/-Sqrt[-24k^2-24k+4m+1])/2
>  now to integer x condition need     -24k^2-24k+4m+1 have to be odd square

The question is WHY such a square EXISTS for a particular value of m?
Maybe there exists such m that -24k^2-24k+4m+1 is never a square?

Regards,
Max


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