Edit: nevermind, didn't think it through.
Edit 2: Here are some thoughts.
p being congruent to 5 mod 6 is the same as saying p is an odd prime congruent to 2 mod 3.
Here is an experiment: Note that
produces all equivalence classes mod 11 besides zero.
If this is true for all odd primes congruent to 2 mod 3 and we can prove it, then the only other piece we'd need is that there exist an infinitude of such primes.