Integer Points on the Elliptic Curve $y^2=x^3+17$. - MathOverflow

Integer Points on the Elliptic Curve $y^2=x^3+17$.

2011-01-23T18:07:45Z

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

Answer by Igor Rivin for Integer Points on the Elliptic Curve $y^2=x^3+17$.

2011-01-23T18:15:08Z

This is addressed in: http://mathoverflow.net/questions/6676/integer-points-of-an-elliptic-curve

Answer by Samir Siksek for Integer Points on the Elliptic Curve $y^2=x^3+17$.

2011-01-23T18:21:18Z

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.

Answer by Gerry Myerson for Integer Points on the Elliptic Curve $y^2=x^3+17$.

2011-01-23T22:29:29Z

Uspensky and Heaslett, 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