We need distinct prime values of k for which there exists a prime p with $p^2 + k < 30$ and $p^2 + k$ prime.

Since $p^2 + k < 30$ and $k \ge 2$, we have $p^2 \le 27 \Rightarrow p \in \{2, 3, 5\}$.

• Case $p = 2$: $p^2 + k = 4 + k < 30 \Rightarrow k \in \{2,3,5,7,11,13,17,19,23\}$. Check primality of $4 + k$: 6 (×), 7 (✓), 9 (×), 11 (✓), 15 (×), 17 (✓), 21 (×), 23 (✓), 27 (×). Valid k: $\{3, 7, 13, 19\}$.

• Case $p = 3$: $p^2 + k = 9 + k < 30 \Rightarrow k \in \{2,3,5,7,11,13,17,19\}$. $9 + k$ is prime only when $k = 2$ (gives 11); for all odd k, $9 + k$ is even > 2 (not prime). Valid k: $\{2\}$.

• Case $p = 5$: $p^2 + k = 25 + k < 30 \Rightarrow k \in \{2,3\}$. 27, 28 → neither is prime. Valid k: $\emptyset$.

Distinct k across all cases: $\{2, 3, 7, 13, 19\} \rightarrow 5$ values.