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.$
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