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This is an attempt to get someone to write a canonical answer, as discussed in this meta thread. We often have people come to us asking for solutions to a diophantine equation which, after some clever manipulation, can be turned into finding rational or integer points on an elliptic curve. See here, here, here, here, here. (This list is biased towards questions I have answered because I remember them best; other people have answered such questions as well.) I would like an answer to which, after one of us has explained the cleverness, we could direct the OP.

An ideal answer would address

  • How to find both rational and integer solutions

  • Good software solutions. Ideally, it would nice to have a walkthrough for doing these things with Sage Notebook, so that people could find solutions without even installing anything.

  • References for how to transform some standard presentations of elliptic curves into Weierstrass form, so that we don't have to write out the algebra every time. I'm thinking of a cubic in $\mathbb{P}^2$ with a rational point that is not a flex, $y^2 = \mbox{degree 4 polynomial}$, a $(1,1)$ curve on $\mathbb{P}^1 \times \mathbb{P}^1$, or an intersection of two quadrics in $\mathbb{P}^3$.

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    Is there an algorithm for finding all the integer points on an elliptic curve?2012-09-27
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    Perhaps make this thread community wiki? Several answerers can address the 3 different points, as I imagine one giant answer from one author would be harder to do.2012-09-27
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    @i.m.soloveichik Both in practice and theory, yes. In theory, yes by a result of Baker; see http://math.stackexchange.com/a/32903/448 . In practice, there are algorithms in SAGE and MAGMA. These are not the same algorithm; my understanding is that the actually implemented methods are correct if they terminate but are not guaranteed to terminate.2012-09-27
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    @RagibZaman Maybe. I had planned to put a bounty on this, because giving a good answer to this deserves a lot of rep. Can I give bounties on CW answers? (I'd be glad to give bounties to multiple users, as appropriate.)2012-09-27
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    Added (reference-request) in light of the last paragraph.2012-10-14

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