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How to show $3X^3+4Y^3+5Z^3$ has no rational solutions ?

I managed to bring it down to the elliptic curve $Y^2=X^3-2^43^360^2$ but now I'm stuck to show group of rational points of this curve is trivial. Please help anyone. Or could any one suggest me some different proof without using elliptic curves ?

dragoboy
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  • There they have directly shown whatever I'm asking. But can't we use any argument related to elliptic curves ? – dragoboy Oct 13 '16 at 04:53
  • How can an expression have a solution? Were you perhaps thinking of an equation? If so, what equation were you thinking of? – JRN Oct 13 '16 at 04:56
  • I wrote the elliptic curve, no ? I want to argue directly from there – dragoboy Oct 13 '16 at 07:58

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