TITLE: Proof of no rational point on Selmer's Curve $3x^3+4y^3+5z^3=0$
QUESTION [21 upvotes]: The 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 $\mathbb Q$, but not in $\mathbb Q$ 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?

REPLY [9 votes]: This problem is in Cassels' book "Local Fields" and I wrote up a solution once along those lines, for an algebraic number theory class.  See 
http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/selmerexample.pdf,
but I should advise that it comes out seeming pretty tedious.  Solutions that involve elliptic curves are more conceptual.  Others have already provided pointers to references for that approach.