Shortest unresolved cubic equations of length l = 12 + log2 3: decide existence of integer solutions

Determine whether each of the following five cubic Diophantine equations has an integer solution, or prove that none exists: (1) y^2 z + y z^2 = x^3 + x^2 + 3x − 1; (2) 2x^3 + 3xy^2 + z^3 + z^2 + 1 = 0; (3) y^2 z + y z^2 = 3x^3 + x^2 + x − 1; (4) (3x − 1) y^2 + x z^2 = x^3 − 2; (5) y^2 z + y z^2 = 6x^3 + x^2 + 1.

Background

The authors enumerate all cubic equations of length l = 12 + log2 3, eliminate those solved by elementary obstructions or exhibiting explicit integer solutions, and resolve the remaining equation 2+4x3+3xy2+z3=0 via cubic reciprocity.

After this resolution, five equations remain as the only cubic equations up to this length for which the solvability problem over the integers is still unresolved. The authors explicitly invite proving nonexistence or finding an explicit solution for each.

References

After equation eq:og is resolved in Theorem \ref{th:og}, the five equations listed in Table \ref{tab:cubshortest} are the only cubic equations of length $l \leq 12+\log_2 3$ for which Problem \ref{prob:cub} remains open. The reader is invited either to find an integer solution to any of these equations or to prove that none exists.

eq:og:

2+4x3+3xy2+z3=0,2+4x^3+3xy^2+z^3=0,

— On the shortest open cubic equations  (2603.29831 - Grechuk et al., 31 Mar 2026) in Section 4 (The new shortest open equations), Table 1 and surrounding text