EDIT, 9 March 2014: when I asked this in 2010, I did not have the courage of my convictions, and so did not ask for an if and only if proof, as Kevin Buzzard quite properly pointed out. Such problems are now somewhat known open problems, as I told them to some experts in 2011. Probably the easiest of the bunch: It is easy to describe a set of integers that are not represented by $4 x^2 + 2 x y + 7 y^2 - z^3;$ I even sent that in as a Monthly problem (December 2010, problem 11539), only one solver, Robin Chapman(December 2012). Open problem: can we prove that the polynomial integrally represents every other number? There is a similar open problem for each discriminant of positive binary quadratic forms with class number three, including the other direction for Kevin Buzzard's answer below.
ORIGINAL: The following problem is my variant of something Irving Kaplansky noticed when we worked together. I do not think it is by nature a difficult problem, it is simply too hard for me to finish.
Suppose we have an integer $C > 0$ that is not divisible by 2 or 3, while there is another integer $F > 0$ such that $ 27 C^2 - 23 F^2 = 4.$
For any integers $x,y,z,$ is it true that $ 2 x^2 + x y + 3 y^2 + z^3 - z \neq C $ and $ 2 x^2 + x y + 3 y^2 + z^3 - z \neq -C ,$ or together $ 2 x^2 + x y + 3 y^2 + z^3 - z \neq \pm C $ ?
I have proved it for the four smallest values of $C,$ those being 1, 599, 14951, 9314449. The case $C=1$ comes directly from the Hudson and Williams paper below. Note that the even values of $C$ fail miserably, they seem to all be values of $ z^3 - z.$ The polynomial $ g(x,y,z) = 2 x^2 + x y + 3 y^2 + z^3 - z $ represents every other number $n$ with $ -10,000,000 \leq n \leq 10,000,000,$ according to my computer.
The Spearman and Williams article (see below) is explicit class field theory, not a topic I know. I should point out that, in retrospect, what I proved for the four smallest (odd) $C$ seems to amount to the statement that $z^3 - z + C$ is irreducible $\pmod q$ for any prime $ q = 2 u^2 + u v + 3 v^2.$
(27 January 2010) I finally got smart and decided to do a part check of the "irreducible" version. For the next three values of $C,$ those being 232488049, 144839681351, 3615189146999, I factored $z^3 - z + C \pmod q$ and found it to be irreducible for primes $q < 1000$ and $ q = 2 u^2 + u v + 3 v^2.$
The two main references are:
Blair K. Spearman and Kenneth S. Williams, "The Cubic Conguence $x^3 + {A} x^2 + {B} x + {C} \equiv 0 \bmod p $ and Binary Quadratic Forms", Journal of the London Mathematical Society, volume 46, 1992, pages 397-410
Richard H. Hudson and Kenneth S. Williams", "Representation of primes by the principal form of discriminant $-{D}$ when the classnumber $h(-{D})$ is $3$", Acta Arithmetica, volume 57, 1991, pages 131-153.
One needs this Lemma: if an integer $n$ has an integer representation as $ n = 2 x^2 + x y + 3 y^2,$ then $n$ is divisible by some prime $ q = 2 u^2 + u v + 3 v^2.$
Everything I know about this problem is in pdf's on
http://zakuski.math.utsa.edu/~jagy/inhom.html
including a proof of the preceding Lemma in jagy_division.pdf and the proof for the four smallest $C$ in jagy_conjecture_23.pdf and a list of intimately related problems in jagy_list.pdf .
I welcome individual responses to this along with posted answers or comments. One of my email addresses can be found using the search feature at