Integer Points on the Elliptic Curve \$y^2=x^3+17\$. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T09:41:27Z http://mathoverflow.net/feeds/question/52979 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52979/integer-points-on-the-elliptic-curve-y2x317 Integer Points on the Elliptic Curve \$y^2=x^3+17\$. Eric Naslund 2011-01-23T18:07:45Z 2013-03-12T00:49:14Z <p>I came across the problem "find all integer solutions to \$y^2=x^3+17\$." </p> <p>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)</p> <p>By numerical calculation I have found that the following integer points \$(x,y)\$ lie on the curve</p> <p>\$(-1,4)\$, \$(-2,3)\$, \$(2,5)\$, \$(4,9)\$, \$(8,23)\$, \$(43,282)\$, \$(52,375)\$, \$(5234,378661)\$ and this is probably all of them.</p> <p>Thanks</p> http://mathoverflow.net/questions/52979/integer-points-on-the-elliptic-curve-y2x317/52982#52982 Answer by Igor Rivin for Integer Points on the Elliptic Curve \$y^2=x^3+17\$. Igor Rivin 2011-01-23T18:15:08Z 2011-01-23T18:15:08Z <p>This is addressed in: <a href="http://mathoverflow.net/questions/6676/integer-points-of-an-elliptic-curve" rel="nofollow">http://mathoverflow.net/questions/6676/integer-points-of-an-elliptic-curve</a></p> http://mathoverflow.net/questions/52979/integer-points-on-the-elliptic-curve-y2x317/52983#52983 Answer by Samir Siksek for Integer Points on the Elliptic Curve \$y^2=x^3+17\$. Samir Siksek 2011-01-23T18:21:18Z 2011-01-23T18:21:18Z <p>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.</p> <p>This method is implemented in several computer algebra packages, including magma. If you type:</p> <p>E:=EllipticCurve([0,0,0,0,17]); IntegralPoints(E);</p> <p>into the online magma calculator at <a href="http://magma.maths.usyd.edu.au/calc/" rel="nofollow">http://magma.maths.usyd.edu.au/calc/</a></p> <p>it will give the eight points you've found already.</p> http://mathoverflow.net/questions/52979/integer-points-on-the-elliptic-curve-y2x317/53002#53002 Answer by Gerry Myerson for Integer Points on the Elliptic Curve \$y^2=x^3+17\$. Gerry Myerson 2011-01-23T22:29:29Z 2011-01-23T22:29:29Z <p>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</p>