The primitive solutions to \(x^3+y^9=z^2\) (Q2565549)

The author proves that the only solutions of the Diophantine equation \(x^3+ y^9= z^2\) with \(x,y,z\in\mathbb{Z}\) and \(\text{gcd}(x,y,z)= 1\) are \[ (x,y,z)= (1,1,0),\,(0,1,\pm 1),\, (1,0\pm 1),\, (2,1,\pm 3),\,(- 7,2,\pm 13). \] First, he reduces this problem to the problem of the determination of all rational points of a finite number of algebraic curves and next he solve it using a Chabauty-like method.