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 find the minimum value of \( p^2 - q^2 \) given that the quadratic equation \( x^2 - px + q = 0 \) has integer solutions, where \( p \) and \( q \) are prime numbers.

### Step 1: Understand the condition for integer solutions
For the quadratic equation \( x^2 - px + q = 0 \) to have integer solutions, the discriminant must be a perfect square. The discriminant \( D \) is given by:

\[ D = b^2 - 4ac \]

Here, \( a = 1 \), \( b = -p \), and \( c = q \). Therefore, we have:

\[ D = (-p)^2 - 4 \cdot 1 \cdot q = p^2 - 4q \]

### Step 2: Set the discriminant as a perfect square
For integer solutions, \( D \) must be a perfect square. Thus, we can write:

\[ p^2 - 4q = k^2 \]

for some integer \( k \). Rearranging gives:

\[ p^2 - k^2 = 4q \]

This can be factored as:

\[ (p - k)(p + k) = 4q \]

### Step 3: Analyze the factors
Since \( q \) is a prime number, \( 4q \) has limited factorizations. The pairs \( (p - k) \) and \( (p + k) \) must be even because their product is even. Let’s denote:

\[ p - k = 2m \quad \text{and} \quad p + k = 2n \]

where \( m \) and \( n \) are integers. Then we have:

\[ (2m)(2n) = 4q \implies mn = q \]

### Step 4: Explore prime pairs
Since \( q \) is prime, the pairs \( (m, n) \) must be such that one of them is 1 (the only way to express a prime as a product of two integers). Thus, we can take:

1. \( m = 1 \) and \( n = q \) or
2. \( m = q \) and \( n = 1 \)

From \( mn = q \), we can derive:

- If \( m = 1 \), then \( n = q \) implies \( p - k = 2 \) and \( p + k = 2q \).
- If \( m = q \), then \( n = 1 \) implies \( p - k = 2q \) and \( p + k = 2 \).

### Step 5: Calculate \( p \) and \( q \)
Let’s consider the first case with the smallest primes:

- Let \( q = 2 \) (the smallest prime).
- Then, \( m = 1 \) gives \( n = 2 \).

From \( p - k = 2 \) and \( p + k = 4 \):

Adding these equations:

\[ 2p = 6 \implies p = 3 \]

Subtracting gives:

\[ 2k = 2 \implies k = 1 \]

### Step 6: Calculate \( p^2 - q^2 \)
Now, substituting \( p = 3 \) and \( q = 2 \):

\[ p^2 - q^2 = 3^2 - 2^2 = 9 - 4 = 5 \]

### Conclusion
The minimum value of \( p^2 - q^2 \) is:

\[ \boxed{5} \]