# Prove that $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ is solvable for all primes p

I am trying to prove that the congruence $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ is solvable for all primes p. I proved it using primitive root, but my professor in number theory told me that it can be more directly done using the hasse-weil theorem in the theory of elliptic curves, but i cant do it. Would someone kindly show me how to make use of the hasse-weil theorem, please? Thank you in advance.

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possibly related –  Old John Nov 10 '13 at 18:24
You did consider $x=y=z=0$ I presume? –  WimC Nov 10 '13 at 19:23
@WimC, no, this is nontrivial solutions. Famous example of Selmer on failure of Hasse-Minkowski once we leave quadratic forms. en.wikipedia.org/wiki/Hasse_principle –  Will Jagy Nov 10 '13 at 20:26
books.google.com/… –  Will Jagy Nov 10 '13 at 20:31
apparently in this: books.google.com/… –  Will Jagy Nov 10 '13 at 20:48