Proof

[Richard Mathar]

Dear Richard,
Yours "countersamples" are congruent to 2 mod 6
7 + 4 (7 k + 7 k2 - x - 2 k x + x2) = 7 + 4 (6 n+2) = 24 n + 15=3(8n+5) 
are divisable by 3 and aren't primes
Best wishes
Artur

Richard Mathar pisze:
> Removing typos from my previous e-mail we have...
>
> aj> From seqfan-owner at ext.jussieu.fr  Wed Apr 23 17:47:33 2008
> aj> Date: Wed, 23 Apr 2008 17:49:24 +0200
> aj> From: Artur <grafix at csl.pl>
> aj> Reply-To: grafix at csl.pl
> aj> To: "N. J. A. Sloane" <njas at research.att.com>
> aj> CC: seqfan <seqfan at ext.jussieu.fr>
> aj> Subject: Proof
> aj> ...
> aj> I hope that I have proof:
> aj> 1) 4 x^2 - 4 x*y + 7 y^2 can be odd prime only  if y is odd
> aj> 4 x^2 - 4 x*(2 k + 1) + 7 (2 k + 1)^2=7 + 28 k + 28 k^2 - 4 x - 8 k x + 
> aj> 4 x^2=
> aj> 7+4(7 k + 7 k^2 - x - 2 k x + x^2)
> aj> 2) Now if we solve equation (7 k + 7 k^2 - x - 2 k x + x^2) = m on 
> aj> variable x and we will be forced x as positive integer
> aj> we are receiving
> aj> x=(1+2k+/-Sqrt[-24k^2-24k+4m+1])/2
> aj> now to integer x condition need     -24k^2-24k+4m+1 have to be odd square
> aj> but these have to be 24w+1 as Zak Seidov states in comment to A001318 
> aj> <http://www.research.att.com/%7Enjas/sequences/A001318>
> ...
>
> This is not clear to me: the -24k^2-24k+4m+1 can be odd squares without
> being of the form 24w+1: they can (at least) also be of the form 24w+9 :
>
> "k=", -4, "m=", 74, "-24k^2-24k+4m+1=", 9, "sqrt(-24*k^2-24*k+4*m+1)=", 3, "mod 24", 8
>
> "k=", -4, "m=", 92, "-24k^2-24k+4m+1=", 81, "sqrt(-24*k^2-24*k+4*m+1)=", 9, "mod 24", 8
>
> "k=", -3, "m=", 38, "-24k^2-24k+4m+1=", 9, "sqrt(-24*k^2-24*k+4*m+1)=", 3, "mod 24", 8
>
> "k=", -3, "m=", 56, "-24k^2-24k+4m+1=", 81, "sqrt(-24*k^2-24*k+4*m+1)=", 9, "mod 24", 8
>
> "k=", -3, "m=", 92, "-24k^2-24k+4m+1=", 225, "sqrt(-24*k^2-24*k+4*m+1)=", 15, "mod 24", 8
>
> "k=", -2, "m=", 14, "-24k^2-24k+4m+1=", 9, "sqrt(-24*k^2-24*k+4*m+1)=", 3, "mod 24", 8
>
> "k=", -2, "m=", 32, "-24k^2-24k+4m+1=", 81, "sqrt(-24*k^2-24*k+4*m+1)=", 9, "mod 24", 8
>
> "k=", -2, "m=", 68, "-24k^2-24k+4m+1=", 225, "sqrt(-24*k^2-24*k+4*m+1)=", 15, "mod 24", 8
>
> "k=", -1, "m=", 2, "-24k^2-24k+4m+1=", 9, "sqrt(-24*k^2-24*k+4*m+1)=", 3, "mod 24", 8
>
> "k=", -1, "m=", 20, "-24k^2-24k+4m+1=", 81, "sqrt(-24*k^2-24*k+4*m+1)=", 9, "mod 24", 8
>
> "k=", -1, "m=", 56, "-24k^2-24k+4m+1=", 225, "sqrt(-24*k^2-24*k+4*m+1)=", 15, "mod 24", 8
>
> "k=", 0, "m=", 2, "-24k^2-24k+4m+1=", 9, "sqrt(-24*k^2-24*k+4*m+1)=", 3, "mod 24", 8
>
> "k=", 0, "m=", 20, "-24k^2-24k+4m+1=", 81, "sqrt(-24*k^2-24*k+4*m+1)=", 9, "mod 24", 8
>
> "k=", 0, "m=", 56, "-24k^2-24k+4m+1=", 225, "sqrt(-24*k^2-24*k+4*m+1)=", 15, "mod 24", 8
>
> "k=", 1, "m=", 14, "-24k^2-24k+4m+1=", 9, "sqrt(-24*k^2-24*k+4*m+1)=", 3, "mod 24", 8
>
> "k=", 1, "m=", 32, "-24k^2-24k+4m+1=", 81, "sqrt(-24*k^2-24*k+4*m+1)=", 9, "mod 24", 8
>
> "k=", 1, "m=", 68, "-24k^2-24k+4m+1=", 225, "sqrt(-24*k^2-24*k+4*m+1)=", 15, "mod 24", 8
>
> "k=", 2, "m=", 38, "-24k^2-24k+4m+1=", 9, "sqrt(-24*k^2-24*k+4*m+1)=", 3, "mod 24", 8
>
> "k=", 2, "m=", 56, "-24k^2-24k+4m+1=", 81, "sqrt(-24*k^2-24*k+4*m+1)=", 9, "mod 24", 8
>
> "k=", 2, "m=", 92, "-24k^2-24k+4m+1=", 225, "sqrt(-24*k^2-24*k+4*m+1)=", 15, "mod 24", 8
>
> "k=", 3, "m=", 74, "-24k^2-24k+4m+1=", 9, "sqrt(-24*k^2-24*k+4*m+1)=", 3, "mod 24", 8
>
> "k=", 3, "m=", 92, "-24k^2-24k+4m+1=", 81, "sqrt(-24*k^2-24*k+4*m+1)=", 9, "mod 24", 8
>
>
> __________ NOD32 Informacje 2701 (20071204) __________
>
> Wiadomosc zostala sprawdzona przez System Antywirusowy NOD32
> http://www.nod32.com lub http://www.nod32.pl 
>
>
>
>