<s>[INST] Integer solutions to $x^3=y^3+2y+1$? 
Find all integral pairs $(x,y)$ satisfying $$ x^3=y^3+2y+1.$$

My approach:
I tried to factorize $x^3-y^3$ as $$(x-y)(x^2 + xy + y^2)=2y+1,$$ but I know this is completely helpless. Please help me in solving this problem.
 [/INST] Or you can write $x=y+z$ for some $z\in Z$ (that is $z=x-y$). So we get a quadratic equation on $y$:
$$ 3y^2z+y(3z^2-2)+z^3-1=0$$
which has a discriminant a perfect square: $$d^2 = (3z^2-2)^2-12z(z^3-1)=-3z^4-12z^2+12z+4$$
So we have $$3z^4+12z^2\leq 12z+4$$
and this can not be true for a lot of integers $z$ (in fact only for $0$ and $1$)...
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