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Here :

https://en.wikipedia.org/wiki/Elliptic_curve#Integral_points

an upper bound for the integral values on an elliptic curve is given.

But this bound is so large that it is of no use in practice, if all integral points have to be found.

How can I find all integral points in practice ?

In the above article, it is mentioned that a "sum" of solutions is again a solution. But I do not think that all solutions can be found this way in general, or is this actually the case ? Moreover, it can be difficult to find one solution or to prove that none exists. Who can help ?

The given answer contains a lot of links, but I would prefer to have either at least the main ideas or how I can determine the solutions with PARI/GP, my favorite calculator-program. Moreover, I noticed that most implemented methods rely on unproved conjectures, if I did not misinterprete it. This would be not really satisfying.

lioness99a
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Peter
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    Actually SAGE has a package to compute all integral points on elliptic curves, but it may not be possible to finish the computation. For more information see also the related question here, and the MO-question here. I like Kevin's answer here:"Finding all the integral points on an elliptic curve is a non-trivial computational problem" ... well, yes. – Dietrich Burde Mar 03 '17 at 10:04
  • @DietrichBurde In other words, we cannot expect to find all the points for every elliptic curve (of course with coefficients of reasonable magnitude) ? If yes, this remark satisfies me. Just formulate it as an answer. – Peter Mar 03 '17 at 10:10

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Although there are many techniques known, and algorithms in computer algebra systems are available (e.g. for SAGE), it is in general not possible to find all integral points on a given elliptic curve in practice. By Siegel's Theorem we know that there are only finitely many integral points on elliptic curves over an algebraic number field, but still this does not help for the question of finding them all. The MO-question has some references from experts, e.g., by Cremona, who has a large database on elliptic curves.

Community
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Dietrich Burde
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  • If we only want to decide whether there is an integral solution; is this always possible in practice ? – Peter Mar 03 '17 at 12:34
  • I think we cannot even decide whether or not there is a rational point on the curve (see Turing machine and Hilbert’s Tenth Problem). – Dietrich Burde Mar 03 '17 at 13:28