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I have seen this equation $y^3 - 3x^2 = p^m$ to determine the solutions. I know this is elliptic curve. I had some knowledge of elliptic curve. But, I was totally upset to determine the solutions of this kind equation. what are the best possible ways to find solutions of such equations? Then I will find solutions of $y^5 - 5x^2 = p^m$, once I know the method to find solutions of $y^3 - 3x^2 = p^m$. Where $p$ is prime and $x, y, m$ are some positive integers.

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I fear that the methods do not easily generalise as you increase the power of $y$. For the case involving $y^2$ you are basically dealing with a Pell equation, and for the cubic, you have an elliptic curve (there are well-established methods for finding solutions of those), but for higher powers, you are really involved with hyperelliptic curves, which are less well understood than elliptic ones, although Falting's theorem proves there are only finitely many rational solutions. – Old John Oct 11 '12 at 11:32
It would appear that you have made no attempt to follow up on the suggestions I made the last time we discussed these equations,… – Gerry Myerson Oct 11 '12 at 12:18
@GerryMyerson! I am really worked on your method and suggestions. I am still failed to see the solutions at $y^3$-3$x^2$ = $p^m$ case. I am sure I can do for n = 4 as well as other numbers. Before that, I am eagerly looking for n = 3 case. I am so sorry, If I am wrong. But, I want to learn. I don't want to leave this problem. – VMRFDU Oct 12 '12 at 4:37
Evidence, VMRFDU, evidence! – Gerry Myerson Oct 12 '12 at 6:10

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