Let $p $ be an odd prime. Assume that we have the following perfect pattern: all the primes below $p$ are successively quadratic residues and quadratic non-residues. What can we say about $p$? Is it possible that only finitely many such $p$ exist?

Edit: Mathematically I mean: denote by $p_n$ the $n$-th smallest prime. Then is it true that there are infinitely many $N$ with the two properties $$1\leq n < N \& \ n \text{ odd } \Rightarrow (\frac{p_n}{p_N})=(\frac{p_1}{p_N})$$ and $$1\leq n < N \& \ n \text{ even } \Rightarrow (\frac{p_n}{p_N})=(\frac{p_2}{p_N}).$$