OFFSET
1,1
COMMENTS
The list on p. 260 of Cox is missing -12, the list in Theorem 7.30 on p. 149 is correct. - Andrew V. Sutherland, Sep 02 2012
Let b(k) be the number of integer solutions of f(x,y) = k, where f(x,y) is the principal binary quadratic form with discriminant d<0 (i.e., f(x,y) = x^2 - (d/4)*y^2 if 4|d, x^2 + x*y + ((1-d)/4)*y^2 otherwise), then this sequence lists |d| such that {b(k)/b(1): k>=1} is multiplicative. See Crossrefs for the actual sequences. - Jianing Song, Nov 20 2019
Proof that {b(k)/b(1): k>=1} is not multiplicative if the form class number for discriminant d is not 1: let f and g be a pair of nonprincipal forms whose composition is principal. By a theorem of Weber (see the W. E. Briggs link), there exists some prime p represented by f and some different prime q represented by g. By Exercise 2.27(a), page 46 of D. A. Cox's book, p and q are not represented by the principal form. But then p*q is represented by the composition of f and g which is principal. Hence we have b(p) = b(q) = 0, but b(p*q) != 0. - Jianing Song, Dec 06 2025
Possible discriminants of nontrivial geometric endomorphism rings of elliptic curves over Q (see the LMFDB link). The geometric endomorphism ring of an elliptic curve E over K is endomorphism ring of the base change of E to an algebraic closure of K. - Jianing Song, Jan 08 2026
REFERENCES
D. A. Cox, Primes of the form x^2+ny^2, Wiley, New York, 1989, pp. 149, 260.
D. E. Flath, Introduction to Number Theory, Wiley-Interscience, 1989.
LINKS
W. E. Briggs, An Elementary Proof of a Theorem About the Representation of Primes by Quadratic Forms, Canadian Journal of Mathematics, 1954;6:353-363.
LMFDB, Elliptic curves over Q.
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
PROG
(PARI) ok(n)={(-n)%4<2 && quadclassunit(-n).no == 1} \\ Andrew Howroyd, Jul 20 2018
CROSSREFS
The sequences {b(k): k>=0}: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), A028609 (d=-11), A033716 (d=-12), A004531 (d=-16), A028641 (d=-19), A138805 (d=-27), A033719 (d=-28), A138811 (d=-43), A318984 (d=-67), A318985 (d=-163).
KEYWORD
fini,full,nonn,nice
AUTHOR
N. J. A. Sloane, May 16 2003
EXTENSIONS
Corrected by David Brink, Dec 29 2007
STATUS
approved
