OFFSET
1,1
COMMENTS
Discriminants of real quadratic fields with 1 class per genus in the form class group quotient by {I,-I}, where I is the principal class. (So I corresponds to the form x^2 - (D/4)*y^2 for 4|D and x^2 - x*y - ((D-1)/4)*y^2 for D == 1 (mod 4), and -I corresponds to the form (D/4)*x^2 - y^2 for 4|D and ((D-1)/4)*x^2 - x*y - y^2 for D == 1 (mod 4)).
Let p be a prime or the additive inverse of a prime. We know that p is represented by a binary quadratic form in the principal genus if and only if kronecker(d,p) >= 0, where d runs through the prime discriminants (fundamental discriminants divisible by only one prime) dividing d. Sequence gives D such that
kronecker(d,p) >= 0 for all d|D prime discriminants => p or -p is represented by the principal form (i.e., there is an element of norm +-p).
Compare A391422, which gives D such that
kronecker(d,p) >= 0 for all d|D prime discriminants => p is represented by the principal form (i.e., there is an element of norm p).
REFERENCES
D. A. Cox, Primes of the form x^2+ny^2, Wiley, New York, 1989.
LINKS
Jianing Song, Table of n, a(n) for n = 1..10000
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
EXAMPLE
24 is a term: we have kronecker(-3,p) >= 0 and kronecker(-8,p) >= 0 <=> p = -2, 3, or p == 1, 19 (mod 24), and every such prime (positive or negative) is represented by the principal form x^2 - 6*y^2. For example, 1^2 - 6*1^2 = -5, 1^2 - 6*2^2 = -23.
136 is a term: for every (positive or negative) prime p such that kronecker(8,p) >= 0 and kronecker(17,p) >= 0, one (and only one) of p or -p is represented by the principal form x^2 - 34*y^2. But since the form class group of discriminant 136 is isomorphic to C_4 (2 classes per genus), we don't know which one is represented. For example, we have 6^2 - 34*1^2 = 2, 17^2 - 34*3^2 = -17, 9^2 - 34*1^2 = 47, and 21^2 - 34*4^2 = -103, but -2, 17, -47, and 103 are represented by the non-principal form 34*x^2 - y^2.
145 is not a term since kronecker(5,5) >= 0 and kronecker(29,5) >= 0, but +-5 are not represented by the principal form. Indeed, they are represented by the non-principal form f(x,y) = 4*x^2 + 9*x*y - 4*y^2 (with f(3,-1) = 5 and f(1,3) = -5), so by Exercise 2.27(a), page 46 of D. A. Cox's book, they are not represented by the principal form.
PROG
(PARI) isA341417(n)={n>1 && isfundamental(n) && !#select(k->k<>2, quadclassunit(n).cyc)}
CROSSREFS
Sequences related to the class groups of real quadratic fields:
| Class groups | Form class groups |
-------------+-------------------------------+-------------------------------+
-------------+-------------------------------+-------------------------------+
For a list of sequences related to the class numbers of real quadratic fields, see A087048.
KEYWORD
nonn
AUTHOR
Jianing Song, Dec 09 2025
STATUS
approved
