Integer solvability of simple Diophantine equations from Grechuk (2022)

Determine whether each of the specific polynomial Diophantine equations presented in Grechuk (2022) admits any integer solutions; that is, for every such equation f(x_1, …, x_k) = 0 listed therein, ascertain whether the solution set over the integers is empty or nonempty.

Background

The paper applies PRAX algorithms to explore approximate emptiness for solution sets of certain Diophantine equations and notes that the literature contains seemingly simple equations whose integral solvability remains unresolved.

As a concrete illustration, the authors discuss Equation (66) in Grechuk (2022), x3 y2 − z3 − 6 = 0, and analyze its solution structure via probabilistic testing, while emphasizing that the broader question of solvability for several such equations is still open.

References

Several simple looking Diophantine equations are considered in , for which it is an open question whether they have any integer solutions.

— Improved Randomized Approximation of Hard Universality and Emptiness Problems  (2403.08707 - Andreou et al., 2024) in Section 6 (Diophantine Equations)