Square Values of Primitive Binary Cubic Forms

Definitions and mathematical statement

An integral binary cubic form is a homogeneous polynomial \[ F(X,Z)=aX^3+bX^2Z+cXZ^2+dZ^3, \qquad a,b,c,d\in{\mathbb{Z}}. \] It is primitive when $\gcd(a,b,c,d)=1$. A primitive representation of an integer $m$ means a solution $F(x,z)=m$ with $x,z\in{\mathbb{Z}}$ and $\gcd(|x|,|z|)=1$. In particular, primitiveness of the representation is a condition on $(x,z)$, not on $(x,y,z)$ or on the represented integer alone.

The question is whether \[ \forall(a,b,c,d)\in{\mathbb{Z}}^4\text{ with }\gcd(a,b,c,d)=1, \quad\exists x,z,y\in{\mathbb{Z}}: \gcd(|x|,|z|)=1, \quad ax^3+bx^2z+cxz^2+dz^3=y^2. \] The pair $(x,z)=(0,0)$ is excluded by coprimality; zero in one coordinate is permitted. As written, $y=0$ is permitted, so a square includes zero. The supplied statement imposes neither irreducibility nor nonzero discriminant, and no such restriction has been added.

The supplied assessment dated 2026-08-21 says the report proves the weaker alternative that a primitive binary cubic form primitively represents a square or twice a square. In symbols, this reported alternative permits \[ F(x,z)=\varepsilon y^2,\qquad \varepsilon\in\{1,2\},\qquad \gcd(|x|,|z|)=1. \] The question asks for the case $\varepsilon=1$ in every instance; the reported alternative does not by itself give that conclusion.

Short English statement

Does every integral cubic form in two variables whose coefficients have no common factor take a square value at some coprime integer pair? The supplied report assessment gives only the weaker square-or-twice-a-square alternative.

Sources

• Local UnsolvedMath registry, permanent ID $30000798$, exact source code catalogue entry, proposed year 2007; no individual proposer is supplied.
• Explicit Methods in Number Theory (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/34, pp. 1957--2034. https://doi.org/10.4171/owr/2007/34.
• Supplied literature reference: Simon, "binary cubic form question," https://ems.press/content/serial-article-files/46118.
• The supplied corrected_verified note reports checking the mathematical content against the official report. The open label and assessment checked 2026-08-21 are supplied dated triage requiring independent verification of status and cited claims.