For prime p sufficiently large, there is always an integer q such that q is a residue mod p, but neither q−1 nor q+1 are; the number of such residues scales like p/8 (and similarly for any sequence of residues/non-residues in three consecutive integers).
What are the best lower bounds on primes p are , for which such "isolated" residues are guaranteed to exist? Do they exist, for instance, for all p ≡ 3 (mod 4) aside from p = 3?
(If this is a typical homework problem, please point me to a textbook for which it is an exercise.)
