If the quadratic equation `x^(2)-px+q=0` where p, q are the prime numbers has integer solutions, then minimum value of `p^(2)-q^(2)` is equal to _________

Answer

To solve the problem, we need to analyze the quadratic equation given and find the minimum value of \( p^2 - q^2 \) where \( p \) and \( q \) are prime numbers and the equation has integer solutions.

### Step-by-Step Solution:

1. **Understanding the Quadratic Equation**:
The quadratic equation is given as:
\[ x^2 - px + q = 0 \]
where \( p \) and \( q \) are prime numbers.

2. **Condition for Integer Solutions**:
For the quadratic equation to have integer solutions, the discriminant must be a perfect square. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac = p^2 - 4q \]
For integer solutions, \( D \) must be a perfect square, say \( k^2 \):
\[ p^2 - 4q = k^2 \]

3. **Rearranging the Equation**:
Rearranging gives us:
\[ p^2 - k^2 = 4q \]
This can be factored as:
\[ (p - k)(p + k) = 4q \]

4. **Finding Prime Values for \( p \) and \( q \)**:
Since \( q \) is a prime number, we can start with the smallest prime numbers for \( q \) and find corresponding values for \( p \).

5. **Testing Values**:
Let's start with \( q = 2 \) (the smallest prime):
\[ (p - k)(p + k) = 4 \times 2 = 8 \]
The pairs of factors of 8 are:
- \( (1, 8) \)
- \( (2, 4) \)

We will test these pairs to find \( p \) and \( k \).

6. **Testing Factor Pair (1, 8)**:
From \( p - k = 1 \) and \( p + k = 8 \):
- Adding these gives \( 2p = 9 \) → \( p = 4.5 \) (not prime)

7. **Testing Factor Pair (2, 4)**:
From \( p - k = 2 \) and \( p + k = 4 \):
- Adding these gives \( 2p = 6 \) → \( p = 3 \) (which is prime)
- Substituting back gives \( k = 1 \).

8. **Calculating \( p^2 - q^2 \)**:
Now we have \( p = 3 \) and \( q = 2 \):
\[ p^2 - q^2 = 3^2 - 2^2 = 9 - 4 = 5 \]

9. **Conclusion**:
Thus, the minimum value of \( p^2 - q^2 \) is:
\[ \boxed{5} \]