A138812 & A138813 Asymptotics

[drew at math.mit.edu]

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Christian G. Bower wrote:
> 
> I don't have the book you're writing about ...

Harary & Palmer?

ja

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>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




... mod initial term, merge?
(at least crossref!)