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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

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    $\begingroup$ This particular example is discussed in Silverman's book The Arithmetic of Elliptic Curves. On the Google books site for that book, do a search for "5234". $\endgroup$
    – KConrad
    Commented Jan 23, 2011 at 20:03
  • $\begingroup$ @KConrad Silverman's book cites the following article by T. Nagell as a reference: Solution de quelque problemes dans la theorie arithmetique des cubiques planes du premier genre. Wid. Akad. Skrifter Oslo I, 1935. Nr. 1. Do you know where can I find that article? I wasn't able to locate it after doing a google search, I also tried looking in the database of my university's library but it doesn't show up in the search. $\endgroup$
    – Adrian Barquero-Sanchez
    Commented Jan 23, 2011 at 20:41
  • $\begingroup$ @Adrian, good luck, those Norwegian journals are not easy to find. Mordell credits Nagell with the result, and gives a different citation: Einige Gleichungen von der Form $ay^2+by+c=dx^3$, Vid Akad Skrifter Oslo 1930, Nr.7. Interlibrary loan might be the way to go. $\endgroup$
    – Gerry Myerson
    Commented Jan 23, 2011 at 23:28
  • $\begingroup$ I haven't seen these volumes myself, only a reference to them: T. Nagell, Collected papers of Trygve Nagell. Vol. 1–4., Edited by Paulo Ribenboim. Queen’s Papers in Pure and Applied Mathematics, Queen’s University 121, Kingston, ON, 2002. $\endgroup$
    – Gerry Myerson
    Commented Jan 24, 2011 at 1:56
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    $\begingroup$ @Gerry Myerson Thanks a lot, I'll try to see if the library has that volume with the collected papers and if not maybe I can try that interlibrary loan you mentioned. $\endgroup$
    – Adrian Barquero-Sanchez
    Commented Jan 24, 2011 at 4:40

3 Answers 3

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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.

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    $\begingroup$ This exactly replicates the first post in the discussion I had pointed to. $\endgroup$
    – Igor Rivin
    Commented Jan 23, 2011 at 18:30
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    $\begingroup$ I liked both of these posts and gave +1 to both. But I feel it is unfair to call this an exact replica. Nowhere in the link is Nigel Smart's book mentioned, or David' bounds and lattice reduction which is why I choose this one. (However, in the link Franz Lemmermeyer post is very useful, so perhaps I should of chose yours) $\endgroup$
    – Eric Naslund
    Commented Jan 23, 2011 at 19:03
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    $\begingroup$ In Sage (sagemath.org), type EllipticCurve([0,0,0,0,17]).integral_points() $\endgroup$
    – William Stein
    Commented Jan 24, 2011 at 1:11
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This is addressed in: Integer points of an elliptic curve

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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

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    $\begingroup$ I learned about this problem from Fröhlich-Taylor: Algebraic Number Theory (Cambridge University Press, 1991), see Section VII.3 there. It would be interesting to see a solution "by hand". $\endgroup$
    – GH from MO
    Commented Jan 23, 2011 at 23:09
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    $\begingroup$ The approach of Delaunay (Delone?) is via reduction to cubic Thue equations of negative discriminant and "should" always work for equations of the shape $y^2=x^3+D$ with $D > 0$. In general, one can always reduce these problems of integer points on a model of an elliptic curve to one of Thue equations, which can subsequently be solved via linear forms in complex logarithms. The advantage of this over appeal to linear forms in elliptic logarithms is that the method is actually an algorithm. I've never done a complexity analysis, but my experience has been that's it's faster too! $\endgroup$
    – Mike Bennett
    Commented Jan 24, 2011 at 1:07
  • $\begingroup$ Wikipedia says, "The spelling Delone is a straightforward transliteration from the Cyrillic alphabet he often used in recent publications, while Delaunay is French language version he used in the early French and German publications." Wikipedia also isn't sure whether he's more famous as a mathematician or as a mountain climber. $\endgroup$
    – Gerry Myerson
    Commented Jan 24, 2011 at 6:27
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    $\begingroup$ @Gerry: There is the following story about Delaunay: at a rather advanced age, he was giving a lecture at Moscow University, and needed to jump up and down to erase the (tall) blackboard. The students tittered, at which point he said "not only are you guys idiots, but I bet you can't do THIS either", at which point he did a handstand (actually, something much harder, but I don't know how to translate into english) on a chair. So, apparently, he kept both the math and the climbing going until the end. $\endgroup$
    – Igor Rivin
    Commented Jan 24, 2011 at 15:42

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