Primes Produced by Quadratic Forms

[Max Alekseyev]

Artur,

There a nice simple characterization of the numbers representable by a
quadratic form:

An integer n is representable as n = a*x^2 + b*x*y + c*y^2 for some
co-prime integers x,y
if and only if the congruence:

z^2 == D (mod 4*|n|)  (where D = b^2 - 4*a*c is the discriminant of
the quadratic form)

has a solution z such that 0 <= z < 2*|n|.

btw, there is another nice, detailed and somewhat elementary book on
quadratic forms by Dedekind:
http://books.google.com/books?id=8h8aWmnp1k8C
Take a look at sections 4 and 5 there (google.books allow to preview
the book by searching inside it)

Regards,
Max

On Fri, Apr 25, 2008 at 1:16 AM, Artur <grafix at csl.pl> wrote:
> Because Sqrt[-12]=2Sqrt[-3] that case wasn't the best
>  A139490(2) is better: x^2+9y^2 and x^2+4xy+y^2
>  Discriminant x^2+9y^2, d= 0^2-4*9=-36  Sqrt[d]=6Sqrt[-1]
>  Discriminant x^2+4xy+y^2, d= 4^2-4*1=-12  Sqrt[d]=2Sqrt[-3]
>  Sets of primes both these are also that same
>  Artur
>
>
>  Artur pisze:
>
>
>
> > Some sequences in ONEIS follow book of Cox (and other authors) suggested
> and prooved that in theory of primes of binary forms ax^2+bxy+cy^2
> discriminant b^2-4ac is the most important.
> > If we take first sample A139490
> <http://www.research.att.com/%7Enjas/sequences/A139490>(1)=A007645
> <http://www.research.att.com/%7Enjas/sequences/A007645>
> > Discriminant of x^2 + xy + y^2 is 1^2-4*1*1= -3
> > but discriminant of x^2 + 3*y^2=0^2-4*1*3= -12
> > Sets of primes of both are that same
> > Artur
> >
> >
> > T. D. Noe pisze:
> >
> > >
> > > > Thus, every prime p such that (-30/p)=1 is produced by one of the
> > > > quadratic forms x^2 + 30y^2, 2x^2 + 15y^2, 3x^2 + 10y^2 or 5x^2 +
> 6y^2. It
> > > > is easy to see (using quadratic reciprocity and its friends) that the
> > > > primes p such that (-30/p)=1 are the primes where p == 1, 11, 13, 17,
> 23,
> > > > 29, 31, 37, 43, 47, 49, 59, 67, 79, 101, or 113 (mod 120).
> > > >
> > > > The first form can, looking modulo 120, only produce primes p == 1,
> 31, 49
> > > > or 79 (mod 120).
> > > > The second form can, looking modulo 120, only produce primes p == 17,
> 47,
> > > > 113 or 23 (mod 120).
> > > > The third form can, looking modulo 120, only produce primes p == 13,
> 43,
> > > > 37 or 67 (mod 120).
> > > > The fourth form can, looking modulo 120, only produce primes p == 11,
> 101,
> > > > 59 or 29 (mod 120).
> > > >
> > > > Since each congruence class modulo 120 of primes p such that (-30/p)=1
> is
> > > > represented only once above, it follows that those congruential
> conditions
> > > > are not only necessary but sufficient for expressibility in any of
> those
> > > > forms.
> > > >
> > > >
> > >
> > >
> > > For those keeping score, these are sequences A033220, A107135, A107136,
> > > A107137.
> > >
> > > Tony
> > >
> > > __________ NOD32 Informacje 2701 (20071204) __________
> > >
> > > Wiadomosc zostala sprawdzona przez System Antywirusowy NOD32
> > > http://www.nod32.com lub http://www.nod32.pl
> > >
> > >
> > >
> > >
> >
> > __________ NOD32 Informacje 2701 (20071204) __________
> >
> > Wiadomosc zostala sprawdzona przez System Antywirusowy NOD32
> > http://www.nod32.com lub http://www.nod32.pl
> >
> >
> >
>