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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](http://en.wikipedia.org/wiki/Algebraic_curve#Curves_of_genus_greater_than_one) proves there are only finitely many rational solutions.2012-10-11
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    It would appear that you have made no attempt to follow up on the suggestions I made the last time we discussed these equations, http://math.stackexchange.com/questions/203834/integer-solutions-to-amnx2-yn-with-various-conditions2012-10-11
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    @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.2012-10-12
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    Evidence, VMRFDU, evidence!2012-10-12

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