Quadratic forms vs congruences

[T. D. Noe]

Dear Tony,

I admit that all this theory is well over my head...

I've spent some days to calculate number "r" of
represntation of "n" as x^2+5y^2 (x,y>=0),
for "n"'s up to ~52,000,000.
To my suprise, the case "r"=19 is absent, while
maximal "r" is ~48 (not remember exactly).

My request(s) to you:
1) do you know why it is so?
2) can you please find the case "r"=19
(and "r=29") (or all cases of r up to say 50.

I could not submit the SEQ without cases of at least
"r"=19
and we do it jointly.
Thanks, zak

PS And i have no access to Cox's book...


--- "T. D. Noe" <noe at sspectra.com> wrote:

> >Perhaps someone has already noticed that
> >
> >A107003 appears to be primes of the form 5+24k
> >A107007 appears to be primes of the form 11+24k and
> 3
> >A107181 appears to be primes of the form 17+24k
> >A107154 appears to be primes of the form 19+24k and
> 3
> >
> >I'm not proving these, but it should be possible.
> 
> Answering my own question...
> 
> The quadratic forms (QFs) in these sequences are
> 
> A107003 5x^2+2xy+5y^2   discriminant=-96
> A107007 3x^2+8y^2       discriminant=-96
> A107181 8x^2+9y^2       discriminant=-288
> A107154 3x^2+16^2       discriminant=-192
> 
> The discriminants are -4N, where N=24, 48, and 72,
> which are all idoneal
> numbers (A000926).  Let N a specific idoneal number.
>  After reading a bit
> of David Cox's "Primes of the form x^2 + n y^2", I
> learned that when the
> discriminant of a QF is -4N, then the primes
> represented by the QF are
> identical to the primes of the form p = s (mod 4N),
> where s is a set of
> numbers.  This is the "one class per genus" case. 
> For the principle QF
> x^2+Ny^2, it is easy computing the set s using
> corollary 2.27.  I did this
> for all idoneal numbers and put the results in
> A139642.  Call s1 the set
> for the principle QF.   For each N there are a total
> of 2^r reduced QFs
> (i.e., the numbers a,b,c are relatively prime) that
> have discriminant N,
> where r is 1, 2, 3, or 4.  It is easy to find those
> reduced QF.  Then for
> each one of those reduced QF, we need to find a set
> s.  Let q be any odd
> prime generated by the reduced QF that does not
> divide N.  Then the set s
> for that QF is merely q*s1 (mod 4N).  It is easy to
> verify
> (computationally) that the s are correct: (1) show
> that they are disjoint
> and (2) show that their union is the set of all
> numbers k such that the
> Jacobi symbol (-k/4N)=1.
> 
> On page 60, Cox lists, for each idoneal N, the
> number of reduced QFs that
> have one class per genus.  The total number of QFs
> is 331.  Quite a few of
> the sequences of primes generated by these 331 QFs
> were in OEIS.  I added a
> congruence formula to each one that did not already
> have a formula.  Then I
> added the remaining sequences.  The main entry is
> A139827, which has a link
> to all 331 sequences.
> 
> Tony
> 
> 



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