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Have the Mordell equation $y^2=x^3+k$ solved for all integer $k$ or not?

Please Could you tell me about a good review papers about such equation.

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  • $\begingroup$ By "solved" you probably mean "integer solutions", not "rational solutions", but both cases are subtle. This distinction should be cleared up. Also, "any $k$" is ambiguous: depending on context, it may mean "all $k$" or it may mean "some $k$" and that ambiguity can confuse non-native English speakers or native English speakers not used to reading math: see math.stackexchange.com/questions/402020/…. You probably mean "all $k$", but still you should write "all" rather than "any" to avoid ambiguity. $\endgroup$
    – KConrad
    Commented Sep 7 at 12:04

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It is not clear what you mean by "completely solved". There are algorithms whose input is $k$ and output is the set of integer solutions. See for example Mordell's equation: a classical approach (the published version is here).

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  • $\begingroup$ I mean for all integer $k$, have they found the solution or not? $\endgroup$
    – Alpha
    Commented Sep 7 at 12:26
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    $\begingroup$ @Alpha By "finding the solutions", mathematicians mean a general procedure, called an algorithm, to find the solutions. In the link I provided, you can find such an algorithm. You give the algorithm a value of $k$, say $k=2024$, and the algorithm gives you all integer solutions $(x,y)$ for that $k$. $\endgroup$
    – GH from MO
    Commented Sep 7 at 12:44

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