Joined: Mon, 29 Dec 2008 17:49:32 UTC Posts: 7573 Location: NCTS/TPE, Taiwan
Indeed, as outermeasure has pointed out, for any x>7, we are left with a linear congruence modulo 7, so we can make it be whatever we feel like by picking x properly.
Indeed, if you want to use mod, you probably need some heavy analytic number theory machinery to help you pick a prime which is 1 (mod 3), for every sufficiently large , for which is not a cubic residue mod . Heuristically such prime should exist for all sufficiently large x
If the cubic residues are uniformly distributed and independent, the probability of fixed x which doesn't have such prime p_x would be about , and so the expected number of such x is finite.
but you get into something very nasty very early on and AFAIK we don't know the behaviour of the cubic Gauss sum to say cubic residues are close enough to randomly distributed for the heuristic argument to work.