Proofs of Conjectures Concerning Entry A033212, the Sequence of Primes Congruent to 1 or 19 (mod 30).

J. B. Tunnell,
Department of Mathematics
The State University of New Jersey
Piscataway, NJ 08854-8019
October 5, 2014

The entry for sequence A033212 contained the conjectures that A033212, the sequence of primes congruent to 1 or 19 (mod 30), was also:
C1. The same as primes of the form x^2-xy+4y^2 (discriminant -15) and x^2-xy+19y^2 (discriminant -75), both with x and y nonnegative;
C2. The same as primes of the form x^2+xy+19y^2 (discriminant -75), with x and y nonnegative; and
C3. The same as primes of the form x^2+5xy-5y^2 (discriminant 45). 

The three conjectures are true : the sequences of prime numbers 
represented by the principal forms of discriminant [-15, -60, -75, 45]
(as was suggested from the first few teerms in the sequences) all agree.

To explain this one needs the notion of the ring class field RC(D)
attached to a discriminant D.  As David A. Cox discusses (for D<0 in 
his book, "Primes of the Form x^2 + n y^2", Wiley, First edition, 1989)
the properties of this field are:

Write D=D0 f^2 for D0 a fundamental discriminant.  Then Q(sqrt(D0)) is
a subfield of RC(D) and

(1).  For each prime number p (prime to D), the ideal (p) splits
    completely in RC(D) if and only if the principal form of
    discriminant D represents the prime number p.  (I'm going to ignore
    primes dividing D in the following.)

(2). The dimension of RC(D) as a Q(sqrt(D))-vector space is the narrow
   class number of the order O(D).  When D=D0 is fundamental, the degree
   is the narrow class number, and the field is the narrow Hilbert class
   field (for D<0, narrow means "usual").

(3). The field RC(D) is a Galois extension over Q, which is abelian
   over Q(sqrt(D)) and is contained in the narrow ray class field of
   conductor f for Q(sqrt(D0)).

(4). RC(D) is contained in the m-th cyclotomic extension of the rationals
   if and only if the list of primes p represented by the principal form
   of discriminant D can be described by congruence conditions on p
   modulo m.

From (1) we see that if RC(D1) \subset RC(D2) then the primes
represented by the principal form of discriminant D2 form a
subsequence of those represented by the principal form of discriminant
D1.  Thus if R(D1)=R(D2) the sequence of primes represented by the two
forms are the same.

There must be some software that computes RC(D) somewhere, but PARI
doesn't seem to have such.  For D<0 (as Cox explains) the computation
of the field involves adjoining values of the j-function.  For D>0 ,
other methods are needed.  But PARI does have the function
quadray(D0,f) which computes the ray class field equation over
Q(sqrt(D0)).  When the various class numbers are small this is often
enough to compute RC(D) by using (3).

As a first example, and to explain the word "narrow" above let's
consider the principal forms x^2+4xy+16y^2 (discriminant -48) and
x^2-3y^2 (discriminant 12).  We can show that they represent the same
prime numbers by computing the ring class fields.  The class number of
Q(sqrt(12)) is 1, the prime ideal (11) splits completely in
Q(sqrt(12)), but looking mod 4 we see that 11 is not represented (-11
is).  So the narrow ring class field must be larger than Q(sqrt(12)).
It is the narrow Hilbert class field, (the maximal abelian extension
unramified at all finite places and ramified at the infinite places).
It is easy to check that Q(sqrt(3),sqrt(-1)) is an abelian extension
of Q(sqrt(12)) which is unramified at all finite places.  Since the
narrow Hilbert field is at most a quadratic extension of the Hilbert class
field, this verifies that RC(12)=Q(sqrt(3),sqrt(-1)).

We can look for other discriminants D with RC(D)=RC(12) by looking at
quadratic subfields in RC(12), say Q(sqrt(-3)).  If we find a
discriminant -3f^2 with RC(-3f^2)=RC(12) we will have an example.
Using PARI it is easy to compute the ray class fields and dimension of
the ring class fields associated to f.  In particular PARI gives

? qfbclassno(-3*16)
%64 = 2

? quadray(-3,4)
%62 = x^2 + 1

Since RC(-3 4^2) is contained in the ray class field, and has the same
degree over Q(sqrt(-3)), RC(-48)=RC(12).  So the forms x^2+4xy+16y^2
and x^2-3y^2 represent the same primes.  Since RC(12) is contained in
the cyclotomic field of 12th roots of unity these are just the primes
congruent to 1 mod 12 (A068228).  This verifies the comments under
that sequence, and verifies that it is the same as A141122.  It also
indicates that the same set of primes can be represented by forms with
discriminants of different signs.

Another small example is x^2+5y^2 of discriminant -20 and x^2+4xy-16y^2
of discriminant 80, where RC(-20)=Hilbert class field of Q(sqrt(5))
                                 =Q(sqrt(-5),sqrt(-1))=RC(80)

Now on to A033212.

Consider the discriminants and related fundamental discriminant factors below
in a short PARI session:

? list
%90 = [-15, -60, -75, 45]
? fund=[-15,-15,-3,5]
%91 = [-15, -15, -3, 5]
? flist=[1,2,5,3]
%92 = [1, 2, 5, 3]
? vector(4,n,qfbclassno(list[n]))
%93 = [2, 2, 2, 1]
? vector(4,n,quadray(fund[n],flist[n]))
%94 = [x^2 - x - 1, x^2 - x - 1, x^4 + x^3 + x^2 + x + 1, x]

For all the negative discriminants in the list this is enough to check that
the ring class field is Q(sqrt(-3),sqrt(-5)) and the prime ideal (p) splits
completely if and only if it does so in the two obvious imaginary subfields,
so p=1 mod 3 and p = 1,4 mod 5. (For discriminant -75 the ray class field
has a unique subfield quadratic over Q(sqrt(-3)) which must be RC(-75)).

Since PARI doesn't have built-in narrow class group computation, we
can try to check that Q(sqrt(-3),sqrt(-5))/Q(sqrt(5) has the right
properties directly.  It is easy to see that it is the narrow ray
class field, since that field is at most quadratic over Q(sqrt(5)) by
line %94 above, and the only ramified prime is 3.  This shows that
RC(45) is either Q(sqrt(5)) or Q(sqrt(5),sqrt(-3)).  There are primes
splitting in the first field which are not represented by the form
x^2+5xy-5y^2 (like 11, although -11 is represented).  Hence
RC(45)=RC(-75)=RC(-60)=RC(-15).

This explains the conjectures in A033212, and probably other
cases when primes represented by one form are a subsequence of those
represented by another.