Title A107006

[N. J. A. Sloane]

so to be cautious I give your definition as a comment,
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Date: Wed, 23 Apr 2008 17:49:24 +0200
From: Artur <grafix at csl.pl>
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Subject: Proof
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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
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> 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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