Find all integers $x$ and $y$ such that $x^3 + y^3 +1 = 6xy$

Question

$x^3 + y^3 +1 = 6xy \quad $ for $x,y \in \mathbb{N}$

Can you give me a hint how to solve this problem, because I can come up with a solution.

Answer 1

WLOG we can assume $x \geq y$. If $x\geq 6$ then we have $$x^3+y^3+1>x^3\geq 6x^2\geq 6xy$$ which is a contradiction,

Hence $x \leq 5$.

Now there are just $5$ cases to check which you can check similarly.

The answer is $(x,y)\in\{(2,3);(3,2)\}$