Bill, This question can be answered at (at least) two levels with very different levels of sophistication.
For point searching on elliptic (and more general curves) there are lots of algorithms and implementatons. There's a stand-alone C program by M. Stoll and other called ratpoints, and my mwrank package which is in Sage (now as part of a larger package just called cremona) contains a C++ wrapper for this, so that Sage can search for rational points on an elliptic curve efficiently. Example: E=EllipticCurve([1,2,3,4,5]) E.point_search(5) Note that the parameter (5 here) is logarithmic, so adding 1 to it increases the running time by a constant factor. There is no option to only give integer points; I will look at possibly adding that. Secondly (and this is much more sophisticated) there are well-developed algorithms to find all (rigorously) the integer points on an elliptic curve by first finding a basis for the Mordell-Weil group (which Sage can already do in many cases, again using mwrank) and then using bounds on linear forms of logarithms combined with lattice reduction. This last part is well described (for example) in Henri Cohen's new book. The only implementations I know of are in SiMath (now defunct) and Magma, but it has been suggested as a good project for a Masters student to reimplement it in Sage, and someone might be doing that. John Cremona On 17/12/2007, William Stein <[EMAIL PROTECTED]> wrote: > > > > - William > > (Sent from my iPhone.) > > Begin forwarded message: > > > From: bill purvis <[EMAIL PROTECTED]> > Date: December 17, 2007 3:05:25 PM MST > To: sage-newbie <[EMAIL PROTECTED]> > Subject: [sage-newbie] Integer points on Elliptic Curves > Reply-To: [EMAIL PROTECTED] > > > > Anyone have any useful sage code to locate integer points on elliptic > curves? I wrote a very crude C program that simply searches over x and y > evaluating the LHS and RHS looking for a match, but that's crude and only > searches a limited range of X/Y. I've not seen any published algorithms > for this, but my experience is still pretty small. Any non-sage > algortihms would also be of interest and I might try my hand at > converting them to sage if there's nothing already available. > > Bill > -- > +---------------------------------------+ > | Bill Purvis, Amateur Mathematician | > | email: [EMAIL PROTECTED] | > | http://bil.members.beeb.net | > +---------------------------------------+ > > > > > > > -- John Cremona --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~----------~----~----~----~------~----~------~--~---
