prove or disprove that the integer solutions to equation $x^2-2xy+y^2-x+y+1=x^3-2y^3$ are only ( x=1, y=0) and (x=-2, y=3).

Question

integer solutions to equation $x^2-2xy+y^2-x+y+1=x^3-2y^3$ are only ( x=1, y=0) and (x=-2, y=3).

This question is related to another question about a system of equations. I showed how these solution can be the solution of system of equation but the condition is that they must be the only solutions.

Answer 1

To disprove the given statement, it is sufficiently enough to provide a counter example;

$(x=-1,y=-1)$ is an integer solution to the given equation.

Hence the statement is false.

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Another way to disprove the statement is that $(-2,3)$ is not a solution.

Hence the statement is false.