What theories do I use to find the integer solutions of $$y^2+2=x^3\ ?$$
Extremely sorry for no work on my part. I could not begin actually.
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See page $7$ of http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/mordelleqn1.pdf – user236182 Dec 30 '15 at 08:44
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This might help you: http://math.stackexchange.com/questions/473180/the-diophantine-equation-x2-2-y3 – Olivier Oloa Dec 30 '15 at 08:45
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http://math.stackexchange.com/questions/1279001/solving-the-mordell-equation-y2-x3-%E2%88%92-2-what-would-be-a-general-strategy – user236182 Dec 30 '15 at 08:45
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1Or see here or here or here or here. – user236182 Dec 30 '15 at 08:55