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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.)

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# Isolated quadratic residues in integers mod p

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 such "isolated" residues 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.)