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I have this Diophantine: $ a^3+a=b^2+1 $ I found $a=2$, $b=3$ works. Also $a=13$ , $b=47$ works. How can I find all the integer solutions?

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    It's just an elliptic curve, and there's$a$lot of theory about it. Try searching for integral points on elliptic curves.2012-08-18

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Using MAGMA and the method described by Pete L. Clark here one finds the complete set of integer solutions is

$(1 , -1), (2 , 3), (13, -47), (1 , 1), (2 , -3), (13 , 47) $

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    Well, I assumed it was a problem in using the software. Thanks for the update.2012-08-19