What numbers are integrally represented by the three-variable inhomogeneous polynomial $$ 7 x^2 + 3 x y + 9 y^2 + z^3 \; \; ? $$ This is one of two similar problems where it is easy to show that certain integers are not represented owing to their prime factorization. The first inkling that it might be possible to prove if and only if on a characterization of represented numbers is due to Kevin Buzzard, in his answer to my first MO question. The list of similar problems for class number three is at LINKSECONDTRY but in any case works in my comment below
My first MO question, with answer by Kevin Buzzard and evidence about why if-and-only-if is difficult, is http://mathoverflow.net/questions/12486/
