Integer points of $y^2 = x^3 - 4x + 9$ I'm trying to find analytically all the integer points of $y^2 = x^3 - 4x + 9$.
The integer solutions I found so far are
$\{-2, 3\}, \{0, 3\}, \{2, 3\}, \{7, 18\}, \{11, 36\}, \{646, 16419\}$
The first five are easy to find with modulo considerations, but how can I find $\{646, 16419\}$? How to prove there are no more solutions?
 A: All integral points are given by
$$
(-2,\pm 3), (0,\pm 3), (2,\pm 3), (7,\pm 18),
(11,\pm 36), (646,\pm 16419)
$$
see the reference for the database in LMFDB.
A: For your first question

,but how can I find $(646,16419)$?

There is an easy method using lines intersecting the curve $y^2 = x^3-4x+9.$
Given any two points on the curve, the unique line
they determine intersects the curve in another point.
In the case of the points $(2,3)$ and $(-2,-3)$
this third point is $(9/4,27/8)$. With this new point
and the point $(2,-3)$ the third point is $(646,16419).$
Your second question

How to prove there are no more solutions?

is much harder. Perhaps descent methods may work here.
A: In SageMath 8.0:
R.<x,y> = QQ[]
E = EllipticCurve(x^3-4*x+9-y^2)
E.integral_points(both_signs=true):

$$\eqalign{&[(-2 : -3 : 1),\cr
 &(-2 : 3 : 1),\cr
 &(0 : -3 : 1),\cr
 &(0 : 3 : 1),\cr
 &(2 : -3 : 1),\cr
 &(2 : 3 : 1),\cr
 &(7 : -18 : 1),\cr
 &(7 : 18 : 1),\cr
 &(11 : -36 : 1),\cr
 &(11 : 36 : 1),\cr
 &(646 : -16419 : 1),\cr
 &(646 : 16419 : 1)]}$$
