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Solve the Mordell equation $x^3+9=y^2$ in integers. It's evident that $(-2,1), (0,3), (3,6), (6,15)$ are solutions, but is there a good way to find all solutions to this equation? I have already tried basic ideas from number theory, such as rewriting it as $(y−3)(y+3)=x^3$. Note that $\gcd(y−3,y+3)=1,2,3,6$. For $1$, I have verified there are no solution. For 2, I have gotten one solution, $(−2,1)$. The case for $3$ is quite difficult. If you consider an elliptic curve, I was looking at aspects of its rank, and how it may lead to a solution.

Martin Sleziak
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Samuel Goodman
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    I'm voting to close this question as off-topic because it was asked a few minutes ago, possibly deleted. There was also an earlier identical question. – Will Jagy Jul 31 '17 at 21:16
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    This identical question contained only a link that doesn't work. It doesn't give a full solution. Can anyone provide a full solution? – Samuel Goodman Jul 31 '17 at 21:17
  • This one I have denoted a favorite, I will be able to find it if deleted. – Will Jagy Jul 31 '17 at 21:17
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    This equation has 10 solutions in the integers. – Michael L. Jul 31 '17 at 21:17
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    E_+00009: r = 1 t = 3 #III = 1 E(Q) = <(-2, 1)> x <(0, 3)> R = 0.8146954406 10 integral points 1. (0, 3) = (0, 3) 2. (0, -3) = -(0, 3) 3. (-2, 1) = 1 * (-2, 1) 4. (-2, -1) = -(-2, 1) 5. (3, 6) = (0, -3) - 1 * (-2, 1) 6. (3, -6) = -(3, 6) 7. (6, 15) = (0, -3) + 1 * (-2, 1) 8. (6, -15) = -(6, 15) 9. (40, 253) = -2 * (-2, 1) 10. (40, -253) = -(40, 253) – Will Jagy Jul 31 '17 at 21:20
  • Is there any full solution or can you just find the solutions via computer? – Samuel Goodman Jul 31 '17 at 21:21
  • Neither one. There is no elementary solution. The site quoted uses elliptic curves. Thus, there is also no elementary computer solution. – Will Jagy Jul 31 '17 at 21:24
  • The link to the site on the other answer doesn't work for me. Could you try reposting the link? – Samuel Goodman Jul 31 '17 at 21:26
  • @Sam FWIW, this graph has points where there is a solution. You can zoom out or scroll to see there is no solution $40<x<3330$. – Χpẘ Jul 31 '17 at 23:36
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    This is a duplicate of https://math.stackexchange.com/q/1199967 . – Gottfried Helms Aug 01 '17 at 00:38

1 Answers1

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$y^2 = x^3+9\tag{1}$
Using online Magma calculator as follows.

$E$:=EllipticCurve($[0, 0, 0, 0, 9]$);
IntegralPoints($E$);

It says that all integral points are $ [ (-2 : -1 : 1), (0 : 3 : 1), (3 : 6 : 1), (6 : -15 : 1), (40 : 253 : 1) ]$.
Hence all integral point are $(x,y)=(-2,\pm1),(0,\pm3),(3,\pm6),(6,\pm15),(40,\pm253).$

Tomita
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