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For each of the following equations, determine whether there are no solutions, finitely many solutions, or infinitely many solutions with $x, y$ justify your answers. $$x^2-5y^2=3 \\\ x^2+7y^2=10000 \\\ x^2-6y^2=30$$

I am not quite sure what I should say here. Every one of those equations should have a non trivial solution if I am not mistaken. Perhaps the first one has no solutions, the second has only finitely many and the last infinitely many?

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closed as off-topic by Servaes, John Omielan, Lord Shark the Unknown, YuiTo Cheng, Cesareo May 9 at 11:43

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    $\begingroup$ Well, you need to consider each one separately. One of them fairly obviously can't have infinitely many solutions, yes? For the others, see if there is an obstruction $\pmod p$ for some prime $p$. $\endgroup$ – lulu May 8 at 11:20
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    $\begingroup$ $x$ and $y$ are integers ? $\endgroup$ – J. W. Tanner May 8 at 13:03
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  1. $3$ is not a quadratic residue mod 5, therefore there are no solutions.
  2. $(100, 0)$ is a solution, solutions are bounded by $x^2 < 10000, 7y^2 < 10000$ thus there at most finitely many solutions.
  3. (6,1) is a solution, there are infinitely many solutions to $x^2-6y^2=1$, thus there are infinitely many solutions.
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  • $\begingroup$ For 2., (x,y)=( $\pm 75$ , $\pm 25)$ are also solutions $\endgroup$ – J. W. Tanner May 8 at 13:04

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