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The projective curve 3x^3+4y^3+5z^3=0 $3x^3+4y^3+5z^3=0$ is often cited as an example (given by Selmer) of a failure of the Hasse Principle: the equation has solutions in any completion of the rationals Q$\mathbb Q$, but not in Q $\mathbb Q$ itself. I don't think I've ever seen a proof of the latter claim - — is someone able to provide an outline? What are the necessary tools?
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Proof of no rational point on Selmer's Curve 3x^3+4y^3+5z^3=0The projective curve 3x^3+4y^3+5z^3=0 is often cited as an example (given by Selmer) of a failure of the Hasse Principle: the equation has solutions in any completion of the rationals Q, but not in Q itself. I don't think I've ever seen a proof of the latter claim - is someone able to provide an outline? What are the necessary tools?
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