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In all my answers so far, I have discussed only the solvability question: does a given equation has any integer solutions or not (in this case the current smallest open equation is still $y(x^3−y)=z^3+3$ of size $H=31$, see Can you solve the listed smallest open Diophantine equations?).

But, of course, finding one solution does not mean to "solve" an equation. In this answer, I consider a much more general problem to decide whether the solution set is finite, and if so, list all the integer solutions. In this formulation, the smallest open equations are $y(z^2-y)=x^3-2$ and $xyz=x^3+y^2+2$ of size $H=22$. Note that for these equations the solvability question has trivial "Yes" answer, but it is open whether the solution sets are finite or infinite, see On the smallest open Diophantine equations: beyond Hilbert's 10 problem.

If the aim is to explicitly describe all solutions (in any form), then the smallest open equations are $y^2+z^2=x^3 \pm 1$ and $y(x^2-y)=z^2-1$ of size $H=17$, see How to describe all integer solutions to $x^2+y^2=z^3+1$? .

Finally, if we aim to describe all solutions in parametric form only, then the smallest open equations are $yz=x^3\pm 1$ and some other equations of size $H=13$, see Find all integer solutions to the following easy-looking Diophantine equations for more details and the full list of equations.

In all my answers so far, I have discussed only the solvability question: does a given equation have any integer solutions or not (in this case the current smallest open equation is still $y(x^3−y)=z^3+3$ of size $H=31$, see Can you solve the listed smallest open Diophantine equations?).

But, of course, finding one solution does not mean to "solve" an equation. In this answer, I consider a much more general problem to decide whether the solution set is finite, and if so, list all the integer solutions. In this formulation, the smallest open equations are $y(z^2-y)=x^3-2$ and $xyz=x^3+y^2+2$ of size $H=22$. Note that for these equations the solvability question has trivial "Yes" answer, but it is open whether the solution sets are finite or infinite, see On the smallest open Diophantine equations: beyond Hilbert's 10 problem.

If the aim is to explicitly describe all solutions (in any form), then the smallest open equations are $y^2+z^2=x^3 \pm 1$ and $y(x^2-y)=z^2-1$ of size $H=17$, see How to describe all integer solutions to $x^2+y^2=z^3+1$? .

Finally, if we aim to describe all solutions in parametric form only, then the smallest open equations are $yz=x^3\pm 1$ and some other equations of size $H=13$, see Find all integer solutions to the following easy-looking Diophantine equations for more details and the full list of equations.

In all my answers so far, I have discussed only the solvability question: does a given equation has any integer solutions or not (in this case the current smallest open equation is still $y(x^3−y)=z^3+3$ of size $H=31$, see Can you solve the listed smallest open Diophantine equations?).

But, of course, finding one solution does not mean to "solve" an equation. In this answer, I consider a much more general problem to decide whether the solution set is finite, and if so, list all the integer solutions. In this formulation, the smallest open equations are $y(z^2-y)=x^3-2$ and $xyz=x^3+y^2+2$ of size $H=22$. Note that for these equations the solvability question has trivial "Yes" answer, but it is open whether the solution sets are finite or infinite, see On the smallest open Diophantine equations: beyond Hilbert's 10 problem.

If the aim is to explicitly describe all solutions (in any form), then the smallest open are equations $y^2+z^2=2x^2\pm 1$, $y^2+z^2=x^3 \pm 1$ and $y(x^2-y)=z^2-1$ of size $H=17$, see How to describe all integer solutions to $x^2+y^2=z^3+1$? .

Finally, if we aim to describe all solutions in parametric form only, then the smallest open equations are $yz=x^3\pm 1$ and some other equations of size $H=13$, see Find all integer solutions to the following easy-looking Diophantine equations for more details and the full list of equations.

In all my answers so far, I have discussed only the solvability question: does a given equation has any integer solutions or not (in this case the current smallest open equation is still $y(x^3−y)=z^3+3$ of size $H=31$, see Can you solve the listed smallest open Diophantine equations?).

But, of course, finding one solution does not mean to "solve" an equation. In this answer, I consider a much more general problem to decide whether the solution set is finite, and if so, list all the integer solutions. In this formulation, the smallest open equations are $y(z^2-y)=x^3-2$ and $xyz=x^3+y^2+2$ of size $H=22$. Note that for these equations the solvability question has trivial "Yes" answer, but it is open whether the solution sets are finite or infinite, see On the smallest open Diophantine equations: beyond Hilbert's 10 problem.

If the aim is to explicitly describe all solutions (in any form), then the smallest open equations are $y^2+z^2=x^3 \pm 1$ and $y(x^2-y)=z^2-1$ of size $H=17$, see How to describe all integer solutions to $x^2+y^2=z^3+1$? .

Finally, if we aim to describe all solutions in parametric form only, then the smallest open equations are $yz=x^3\pm 1$ and some other equations of size $H=13$, see Find all integer solutions to the following easy-looking Diophantine equations for more details and the full list of equations.

In all my answers so far, I have discussed only the solvability question: does a given equation has any integer solutions or not (in this case the current smallest open equation is still $y(x^3−y)=z^3+3$ of size $H=31$, see Can you solve the listed smallest open Diophantine equations?).

But, of course, finding one solution does not mean to "solve" an equation. In this answer, I consider a much more general problem to decide whether the solution set is finite, and if so, list all the integer solutions. In this formulation, the smallest open equations are $y(z^2-y)=x^3-2$ and $xyz=x^3+y^2+2$ of size $H=22$. Note that for these equations the solvability question has trivial "Yes" answer, but it is open whether the solution sets are finite or infinite, see On the smallest open Diophantine equations: beyond Hilbert's 10 problem.

Finally, if we aim for parametric description of all solutions, then the smallest open equations are $yz=x^2+x\pm 1$ of size $H=11$. For these equations, it is trivial that there are infinitely many integer solutions, but it is not clear if we can write them all in parametric form, see Polynomial parametrization of the solutions to $yz=x^2+x\pm 1$.

In all my answers so far, I have discussed only the solvability question: does a given equation has any integer solutions or not (in this case the current smallest open equation is still $y(x^3−y)=z^3+3$ of size $H=31$, see Can you solve the listed smallest open Diophantine equations?).

But, of course, finding one solution does not mean to "solve" an equation. In this answer, I consider a much more general problem to decide whether the solution set is finite, and if so, list all the integer solutions. In this formulation, the smallest open equations are $y(z^2-y)=x^3-2$ and $xyz=x^3+y^2+2$ of size $H=22$. Note that for these equations the solvability question has trivial "Yes" answer, but it is open whether the solution sets are finite or infinite, see On the smallest open Diophantine equations: beyond Hilbert's 10 problem.

If the aim is to explicitly describe all solutions (in any form), then the smallest open are equations $y^2+z^2=2x^2\pm 1$, $y^2+z^2=x^3 \pm 1$ and $y(x^2-y)=z^2-1$ of size $H=17$, see How to describe all integer solutions to $x^2+y^2=z^3+1$? .

Finally, if we aim to describe all solutions in parametric form only, then the smallest open equations are $yz=x^3\pm 1$ and some other equations of size $H=13$, see Find all integer solutions to the following easy-looking Diophantine equations for more details and the full list of equations.