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The equation $y^2 = x^3 + k$ for $k = (4n-1)^3 - 4m^2$, with $m, n \in \mathbb{N}$ and no prime number that p is congruent to 1 modulo 4 divids m, doesn't have any answer and its proof can be obtained by using quadratic reciprocity law.

Do you know answers of this equation for two or three different values of $k$? In addition, do you know any reference about that?

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    $\begingroup$ This question could sure use a little effort on the format aspect (sentence structure, LaTex, etc). $\endgroup$ Commented Jan 16, 2015 at 20:34
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    $\begingroup$ I tried to edit your question, but some passages were too unclear to attempt an edit. Could you clarify the question? $\endgroup$ Commented Jan 16, 2015 at 20:37
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    $\begingroup$ The abrupt mention of $p$ needs clarification. Is $p$ supposed to be $k$, or a prime factor of $k$? $\endgroup$ Commented Jan 16, 2015 at 20:39
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    $\begingroup$ I wish more people knew what they were missing out on by not knowing how to TeX! $\endgroup$ Commented Jan 16, 2015 at 20:41
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    $\begingroup$ "As you know": that is not a good way to start, as many who read the question will not know. I think "count $m$" is supposed to be "divides $m$." $\endgroup$ Commented Jan 16, 2015 at 20:47

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This is a famous class of elliptic curves, called Mordell's equation, or sometimes Mordell-Bachet equation. See also here, or here for some discussions on MSE. For a specific example with $k=2000000$ see also here. A further reference is this article by Keith Conrad.

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