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I recently read http://cp4space.wordpress.com/2013/11/25/crash-course-in-gaussian-integers/, where it teaches you how to solve some diophantine equations of the form x^2+n=y^3 using the gaussian integers, and I have figured out the solutions to x^2+n=y^3 for n=1, 2, 3, and 4, but for higher numbers, like 5, Z(sqrt(-n)) does not have the fundamental theorem of arithmetic, so you can't solve the equation in the same way.

So my question is: How do you solve equations like x^2+5=y^3?

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    $\begingroup$ For $n = 5$ you can use reduction mod 4. See Theorem 2.2 of math.uconn.edu/~kconrad/blurbs/gradnumthy/mordelleqn1.pdf. $\endgroup$
    – KConrad
    Commented Dec 13, 2013 at 5:58
  • $\begingroup$ I don't oppose closing this question, but to say that "How do you solve equations like $x^2+5=y^3$?" "does not appear to be about research level mathematics" is quite absurd. $\endgroup$
    – Gerry Myerson
    Commented Dec 13, 2013 at 21:47

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It depends on $n$. For some values, simple congruences will do. For others, the quadratic field is the way to go --- even if there is no unique factorization, sometimes if the class number is right you can still get to a conclusion. For some $n$, you want to work in the cubic field with $\root3\of n$. One way to get some idea of what methods are of use is to type "Mordell equation" into a search engine, and see what shows up.

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