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Description: An elliptic curve is defined as $y^2=x^3+ax^2+b$, here, $a, b$ are integers and $(x',y')$ is the point on curve with smallest possible integers coordinate of the elliptic curve.

Question: Is there any inequality regarding integer coefficient $a, b$ and the $x',y'$?

Michael
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  • $x',y'$ needn't exist. – Wojowu Jan 13 '20 at 21:10
  • @Wojowu what is the necessary and sufficient condition for the existence? – Michael Jan 13 '20 at 21:13
  • I doubt any sensible criteria in terms of $a,b$ exist. The only thing that comes to my mind is that if the curve has a nontrivial torsion point, then by Nagell-Lutz it has integer coordinates. – Wojowu Jan 13 '20 at 21:17
  • @Wojowu the curve I am working on, atleast has an integer solution, can we infer the there are other integer solutions? – Michael Jan 13 '20 at 21:21
  • That said, if we assume there does exist a point on the elliptic curve, you can get an explicit upper bound from a result of Baker mentioned here – Wojowu Jan 13 '20 at 21:21
  • If $(x,y)$ is an integer solution and $y\neq 0$, then $(x,-y)$ is another solution. You cannot conclude there are other ones - there are curves with only two integer points like that. In fact, an elliptic curve will always have finitely many points. – Wojowu Jan 13 '20 at 21:22
  • @Wojowu thanks for your help. – Michael Jan 13 '20 at 21:29

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