TITLE: Integer Points on the Elliptic Curve $y^2=x^3+17$.
QUESTION [13 upvotes]: 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

REPLY [7 votes]: Uspensky and Heaslet, Elementary Number Theory, published in 1939, credits Delaunay (on page 400) with showing that $y^2=x^3+17$ has only the eight solutions, and goes on to say, "Whether his method will always work is still an open question, and the problem, despite its simple appearance, is a very difficult one." No reference is cited