Title A107006

[Maximilian Hasler]

What one can say is e.g.

%C A107006
Identical to the list of primes = 7 (mod 24) at least for all 9824
terms not exceeding 10^6.
Since 4x^2-4xy+7y^2 = (2x-y)^2+6y^2 (x,y >= 0) can be prime only for
odd y, the set can also be written as
{ primes of the form (2k+1)^2+6(2m+1)^2 ; k,m>=0 }.
%H A107006 M. F. Hasler, <a
href="http://www.research.att.com/~njas/sequences/b107006.txt">
   Table of n, a(n) for n=1,...,9824</a>.
%o A107006 (PARI)
A107006(Nmax) = { local(S=[],t); forstep(y=1,sqrtint(Nmax\6),2,
t=6*y^2; forstep(xx=1,sqrtint(Nmax-t),2, isprime(t+xx^2)|next;
S=setunion(S,[t+xx^2]))); vecsort(eval(S)) }
%o A107006 /* create list up to 10^6 */ A107006(10^6);
%o A107006 /* check equality with primes = 7 mod 24 */
c=0;p=1;while(c<#%,while((p=nextprime(p+1))% 24-7,); p==%[c++] & next;
error([c,%[c],p]))

Maximilian


On Wed, Apr 23, 2008 at 10:22 AM, N. J. A. Sloane <njas at research.att.com> wrote:
> Artur,  Thanks for this comment.  Let me explain what I did and why.
>  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!
>  In this case you may well have a proof, of course - do you?
>
>  >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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