It came up when I was trying to solve the equality $\sum_{i = 1}^{x}i^2 =n^2$ for integers $x$ and $i$. I've reduced it to the equation $2x^3+3x^2+x-6n^2 = 0$, which I don't know how to tackle. Is there some sort of method for solving a diophantine equation like this?
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you should learn elliptic curve and you can solve this question in minutes. – DeepSea Jan 25 '15 at 02:22
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@Back2Basic, could you show me how, and maybe give me a good resource to start with? – recursive recursion Jan 25 '15 at 02:24
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There is just the one solution. Pretty famous episode, related to the Leech lattice, or rather, one way of working on it. Apparently page 258 in Mordell's book. – Will Jagy Jan 25 '15 at 02:26
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Yeah, Mordell says it is difficult, G. N. Watson gave a complicated answer in 1919, Ljunggren in 1952 by a Pell equation in a quadratic field. Mordell calls it the problem of Lucas on the square pyramid. Why are you considering this? Oh, just one nontrivial solution: the others are $x=0,1,-1.$ – Will Jagy Jan 25 '15 at 02:36
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Just out of curiosity. Could you post the links to these solutions you're finding? Maybe I can understand one of them. – recursive recursion Jan 25 '15 at 02:37
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Mordell, Diophantine Equations (1969), Wolfgang Ebeling, Lattices and Codes, 2nd edition (2002). – Will Jagy Jan 25 '15 at 02:40
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Thanks. I know the solution thanks to the magic of wolfram, I just wanted to know how to get there. – recursive recursion Jan 25 '15 at 02:42
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Alrighty; some elementary solutions were found in the 1980's. Henri Cohen summarizes these on pages 424 to 427 of
Number Theory: Volume I: Tools and Diophantine Equations,
see
Will Jagy
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@Lucian, no, found it, see http://matwbn.icm.edu.pl/ksiazki/aa/aa86/aa8631.pdf – Will Jagy Jan 25 '15 at 18:47