I came across the problem "find all integer solutions to $y^2=x^3+17$." 


I've tried several things, without any success, and I was hoping that someone could help out. (Some ideas or a reference for where to find it are both appreciated)


By numerical calculation I have found that the following integer points $(x,y)$ lie on the curve


$(-1,4)$, $(-2,3)$, $(2,5)$, $(4,9)$, $(8,23)$, $(43,282)$, $(52,375)$, $(5234,378661)$ 
and this is probably all of them.


Thanks



Integer Points on the Elliptic Curve $y^2=x^3+17$.
There is a standard method for computing all integral points on an elliptic curve using David's bounds and lattice reduction. The method can be found in the book:
Nigel Smart, "The Algorithmic Resolution of Diophantine Equations", Cambridge University Press.


This method is implemented in several computer algebra packages, including magma. If you type:


E:=EllipticCurve([0,0,0,0,17]);
IntegralPoints(E);


into the online magma calculator at <http://magma.maths.usyd.edu.au/calc/>


it will give the eight points you've found already.


