TITLE: Three consecutive quadratic residues problem
QUESTION [8 upvotes]: Prove that doesn't exist $N\in\mathbb{N}$ with property: for all primes $p>N$ exist $n\in\{3, 4,\ldots, N\}$ such that $n, n-1, n-2$ are quadratic residues modulo $p$.

REPLY [7 votes]: This is Theorem 2 in D. Lehmer & E. Lehmer, On runs of residues,
Proceedings of the American Mathematical Society, Vol. 13, No. 1 (Feb., 1962), 102-106.
The proof there uses quadratic reciprocity and Dirichlet's Theorem on primes in arithmetic progression. They also deal with similar problems for k-th powers.