Proof

[Graeme McRae]

A simpler proof...  I considered the 576 possible combinations of x,y (mod 
24), and found that possible values of 4x^2-4xy+7y^2 are 0, 4, 7, 12, 15, 
and 16.  Of these, only 7 (mod 24) can be prime.

----- Original Message ----- 
From: "Artur" <grafix at csl.pl>
To: "N. J. A. Sloane" <njas at research.att.com>
Cc: "seqfan" <seqfan at ext.jussieu.fr>
Sent: Wednesday, April 23, 2008 8:49 AM
Subject: Proof


> Dear Neil,
> I hope that I have proof:
> 1) 4 x^2 - 4 x*y + 7 y^2 can be odd prime only  if y is odd
> 4 x^2 - 4 x*(2 k + 1) + 7 (2 k + 1)^2=7 + 28 k + 28 k^2 - 4 x - 8 k x + 4 
> x^2=
> 7+4(7 k + 7 k^2 - x - 2 k x + x^2)
> 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
> but these have to be 24w+1 as Zak Seidov states in comment to A001318 
> <http://www.research.att.com/%7Enjas/sequences/A001318>
> from these condition m=6n and 24n>=24k^2+24k
> finally
> 7+4(7 k + 7 k^2 - x - 2 k x + x^2)=7+4(6n)=24n+7 q.e.d.
> Best wishes
> Artur
>
>
>
>
>
>
>
> N. J. A. Sloane pisze:
>> Artur,  Thanks for this comment.  Let me explain what I did and why.
>> Reply-To: njas at research.att.com
>> X-Mailer: mailx (AT&T/BSD) 9.9 2008-02-12
>> Mime-Version: 1.0
>> Content-Type: text/plain; charset=us-ascii
>> Content-Transfer-Encoding: 7bit
>> To: grafix at csl.pl, noe at sspectra.com,
>>     seqfan at ext.jussieu.fr
>> Subject: Re: Title A107006
>> Cc: njas at research.att.com
>>
>> I KNOW the definition is  Primes of the form 4x^2-4xy+7y^2, with x and y 
>> nonnegative
>>
>> You tell me this is the same as  Primes of the form 24n+7.
>>
>> But do you have a proof?  This subject is very tricky, as you know.
>> Have you studied the book by Cox ?
>>
>> I am not CERTAIN you have a proof that they are the same,
>> so to be cautious I give your definition as a comment,
>> with your name attached, in case it is wrong!
>>
>> You see, I get a lot of comments from amateurs, who
>> think that because two sequences agree for 40 terms,
>> they are the same.  I cannot stop and ask each person,
>> "do you have a proof, or is what you tell me just a guess?"
>>
>> In fact, many of the people who use the OEIS don't even know what a proof 
>> is!
>>
>> In this case you may well have a proof, of course - do you?
>>
>> Best regards
>>
>> Neil
>>
>>
>>> Date: Wed, 23 Apr 2008 12:06:59 +0200
>>> From: Artur <grafix at csl.pl>
>>> Reply-To: grafix at csl.pl
>>> To: seqfan <seqfan at ext.jussieu.fr>, noe at sspectra.com
>>> Subject: Title A107006
>>>
>>
>>
>>> In my private opinion the title of A107006 
>>> <http://www.research.att.com/%7Enjas/sequences/A107006> would be better 
>>> if it was changed to the simpler version: "Primes of the form 24k+7"
>>> Artur
>>>
>>
>>
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