Timeline for How to prove there are exactly $8$ integer points on the elliptic curve $y^2 = x^3 + 17$
Current License: CC BY-SA 4.0
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| Apr 13, 2021 at 6:14 | comment | added | Milo Moses | @NoamD.Elkies That is a very interesting point. I have not heard about Skolem's $p$-adic method before. Thank you for adding this to the discussion. | |
| Apr 13, 2021 at 2:41 | comment | added | Noam D. Elkies | I can write an outline of how to reduce $y^2 = x^3 + D$ to a finite list of cubic Thue equations (equations $P(a,b) = d$ for given $d \neq 0$ and irreducible homogeneous $P$ of degree $3$). When as here $D>0$, each $P$ has one real root and one conjugate pair of complex roots, so the relevant unit group is of rank $1$ and one can finish off with Skolem's $p$-adic method rather than cite Baker's bounds. That's still not entirely elementary, but much more accessible than invoking Baker (and then doing the additional work of efficiently searching up to the huge Baker bounds). | |
| Apr 13, 2021 at 2:35 | comment | added | Noam D. Elkies | Cody K. has a legitimate question: Mordell wrote his book in 1969, shortly after Baker gave the first effective bounds but many decades before we had the software and the sufficiently fast computers to answer such questions routinely. Chris Wuthrich already linked to MO Question 52979, where Gerry Myerson reports that "Mordell credits Nagell with the result" and cites a 1930 paper, and also cites another source crediting Delaunay. The technique is not entirely elementary but needs only algebraic number theory in some quadratic and cubic fields. [cont'd] | |
| Apr 13, 2021 at 0:02 | comment | added | Milo Moses | @CodyK. What I'm saying is that you can use an effective version of Stark's theorem, or whatever algorithm Sage is implementing, to find a proof that these are the only points. I do not think that Mordell has some special trick for this curve. He just used a computer with the indicated methods and these were the only points found in his search. | |
| Apr 12, 2021 at 22:26 | comment | added | Cody K. | Thank you for your answer. I know those 8 points already. But I want to know the formal proof that those are the only integer points on that curve. I have read "Diophantine Equations" of L.J.Mordell which he stated on page 246 that when D =17, there are exactly 8 solutions with the list of solutions followed. But he did not provide the proof of that, which I am looking for. | |
| Apr 12, 2021 at 19:50 | history | edited | Milo Moses | CC BY-SA 4.0 |
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| Apr 12, 2021 at 19:25 | history | edited | Milo Moses | CC BY-SA 4.0 |
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| Apr 12, 2021 at 19:18 | history | answered | Milo Moses | CC BY-SA 4.0 |