Consider the elliptic curve $y^2 = x^3 + 17$. I know that there are exactly $8$ integer points $(x,y)$ with $y>0$. But how do I prove it? Is there any specific approach to it or any proof for it?
How to prove there are exactly $8$ integer points on the elliptic curve $y^2 = x^3 + 17$ [duplicate]
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2$\begingroup$ Welcome to Math Overflow! If you think that an answer fully resolves your question, it is customary to accept the answer by hitting the green checkmark under the upvote/downvote buttons. $\endgroup$– Milo MosesCommented Apr 12, 2021 at 19:53
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$\begingroup$ Franz Lemmermayer's answer here mathoverflow.net/questions/6676/… may be helpful. I find the short explanation in Smart's "The algorithmic resolution of diophantine equations" (XIII.3) well explained. $\endgroup$– Chris WuthrichCommented Apr 12, 2021 at 23:39
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Using the mathematical programming language Sage, we can run the E.integral_points() command to get a (proof verified) confirmation that there are only $8$ integral points.
The points are $(-2,3)$, $(-1,4)$, $(2,5)$, $(4,9)$, $(8,23)$, $(43, 282)$, $(52, 375)$, and $(5234,378661)$. Note that some would call this $16$ points, since all of these points remain solutions to your curve when $(x,y)$ is replaced with $(x,-y)$.
If you wish to learn more about the algorithm implemented by Sage, I do believe that they are transparent about their algorithms on their website.
NOTE: This case is much easier than the general one, since there is a theorem of Stark (see Theorem 7.2 in Silverman's AEC) which states that there is an effectively computable constant $C_{\epsilon}$ for every $\epsilon$ such that all integral solutions of the curve $y^2=x^3+D$ satisfy $\log(\max\{x,y\})\leq C_{\epsilon}|D|^{1+\epsilon}$, so for $D=17$ there are really not too many values to check.
The Hall-Lang conjecture states that this bound can be reduced to one of the form $|x|\leq C_{\epsilon}D^{2+\epsilon}$ but seeing as that is still a conjecture it cannot be used for finding rational points on your curve.
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$\begingroup$ 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. $\endgroup$– Cody K.Commented Apr 12, 2021 at 22:26
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$\begingroup$ @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. $\endgroup$ Commented Apr 13, 2021 at 0:02
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6$\begingroup$ 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] $\endgroup$ Commented Apr 13, 2021 at 2:35
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8$\begingroup$ 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). $\endgroup$ Commented Apr 13, 2021 at 2:41
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1$\begingroup$ @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. $\endgroup$ Commented Apr 13, 2021 at 6:14