TITLE: Is every number the sum of two cubes modulo p where p is a prime not equal to 7?
QUESTION [21 upvotes]: If p is a prime other than 7, can every integer be written as sum of two cubes modulo p?
Has Waring's problem mod p for cubes been proved simply and directly? 
Thanks for your proof.
Lemi

REPLY [2 votes]: The argument in this MO answer: Simple proofs for the existence of elliptic curves having a given number of points
shows that the number of solutions to $y^2=f(x)$, $f$ a cubic, is at most $3p/2$. Taking a quadratic twist, it follows that the number of solutions is at least $p/2$. The equation $x^3+y^3=a$ can be put in Weierstrass form with a slightly messy but elementary calculation and the result follows.