Integer points of $y^2 = x^3 - 4x + 9$

Question

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?

Answer 1

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.

Answer 2

In SageMath 8.0:

R.<x,y> = QQ[]
E = EllipticCurve(x^3-4*x+9-y^2)
E.integral_points(both_signs=true):

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

Answer 3

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.