2021/032) Find non -ve integer x and y satisfying $x^3 = y^3 + 2y + 1$

We have $(y+2)^3 = y^3 + 6y^2+12y + 8 > y^3 + 2y + 1$

and $y^3 < y^3 + 2y + 1$

so x = y + 1

or $(y+1)^3 = y^2 + 3y^2 + 3y + 1 = y^2 + 2y + 1$

or  $2y^2 + y = 0$ giving y = 0(other root -ve and so not admissible)  and x = 1