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$S$ denotes the set of rational points of any curve in the plane.

What is more amazing between a) and b)?

a) $S$ is dense in the curve $y^2=x^3-2^4\cdot3^3\cdot7^2$

b) $S=\emptyset$ in the curve $y^2=x^3-2^4\cdot3^3\cdot5^2$

N.B.- Both, a) and b) are true.

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Piquito
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  • Do you mean $S$ is a function of the curve or do you mean $S$ is the totality of all rational points on all curves (presumably defined over $\Bbb Q$)? – Gregory Grant Apr 12 '16 at 16:10
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    I assume you mean curve (a) has infinitely many rational points and (b) has none. Well it always has at least one doesn't it? The point at infinity is rational isn't it? Anyway neither is more amazing. The rational points form a group, sometimes that group is infinite sometimes it is finite. The fact that they form a group at all is what is amazing. – Gregory Grant Apr 12 '16 at 16:13
  • @GregoryGrant: I have taken the non-homogenized curve, i.e, the "rational" point at infinity is excluded. On the other hand infinitely many is weaker than dense. You are right with your irony or sarcasm about the group. I have edited deleting "elliptic". Regards. – Piquito Apr 12 '16 at 16:34
  • @GregoryGrant: Can you prove a)?. Thanks. – Piquito Apr 12 '16 at 16:36

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