<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] we first consider only $x> y\ge0$
suppose $x=y+a$
then the equation is $(y+a)^3=y^3+2y+1$
$y^3+3ay^2+3a^2y+a=y^3+2y+1$
$3ay^2+(3a^2-2)y+(a-1)=0$
however, as $a\ge1$, $3a^2-2\ge0$,$a-1\ge0$
If $y>0$, then $3ay^2+(3a^2-2)y+(a-1)>0$, so $y=0$
then, $x^3=1$
$(x,y)=(1,0)$ are only integral solution
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