Find the least positive integer $n$ such that for every prime number $p, p^2 + n$ is never prime.
1. **Understanding the problem**: We need to find the smallest positive integer \( n \) such that for every prime number \( p \), the expression \( p^2 + n \) is never a prime number.

2. **Initial consideration**: Let's consider the nature of \( p^2 \). For any prime \( p \), \( p^2 \) is always an odd number except when \( p = 2 \). Therefore, \( p^2 + n \) will be odd if \( n \) is even, and it will be even if \( n \) is odd.

3. **Case analysis**:
   - If \( n \) is even, then \( p^2 + n \) will be odd for any odd prime \( p \). However, we need to ensure that \( p^2 + n \) is never prime for any prime \( p \), including \( p = 2 \).
   - If \( n \) is odd, then \( p^2 + n \) will be even for any odd prime \( p \). Since the only even prime is 2, \( p^2 + n \) will be composite if \( p \neq 2 \).

4. **Checking specific values**:
   - For \( p = 2 \), \( p^2 = 4 \). We need \( 4 + n \) to be composite.
   - If \( n = 1 \), then \( 4 + 1 = 5 \), which is prime.
   - If \( n = 2 \), then \( 4 + 2 = 6 \), which is composite.
   - If \( n = 3 \), then \( 4 + 3 = 7 \), which is prime.
   - If \( n = 4 \), then \( 4 + 4 = 8 \), which is composite.
   - If \( n = 5 \), then \( 4 + 5 = 9 \), which is composite.

5. **Verification for all primes**:
   - For \( n = 2 \), \( p^2 + 2 \) must be checked for all primes \( p \):
     - For \( p = 2 \), \( 2^2 + 2 = 6 \), which is composite.
     - For \( p = 3 \), \( 3^2 + 2 = 11 \), which is prime.
   - For \( n = 4 \), \( p^2 + 4 \) must be checked for all primes \( p \):
     - For \( p = 2 \), \( 2^2 + 4 = 8 \), which is composite.
     - For \( p = 3 \), \( 3^2 + 4 = 13 \), which is prime.
   - For \( n = 5 \), \( p^2 + 5 \) must be checked for all primes \( p \):
     - For \( p = 2 \), \( 2^2 + 5 = 9 \), which is composite.
     - For \( p = 3 \), \( 3^2 + 5 = 14 \), which is composite.
     - For \( p = 5 \), \( 5^2 + 5 = 30 \), which is composite.
     - For \( p = 7 \), \( 7^2 + 5 = 54 \), which is composite.

6. **Conclusion**: The smallest \( n \) such that \( p^2 + n \) is never prime for any prime \( p \) is \( n = 5 \).

The final answer is \( \boxed{5} \).