Given the equation:

$(p + q)^p = (q - p)^{2p - 1}$

where $p$ and $q$ are prime numbers, we want to find the value of $pq$.

Step 1: Analyzing the equation

Let's start by analyzing the equation given:

$(p + q)^p = (q - p)^{2p - 1}$

Since $p$ and $q$ are primes, let's consider different cases to understand how the expression behaves. Notice that for the right-hand side $(q - p)^{2p - 1}$ to be defined and meaningful, $q - p$ must be positive (i.e., $q > p$).

Step 2: Trying specific small prime values

Let's try the smallest prime number $p = 2$ and check the behavior of the equation for different prime $q$.

Case 1: $p = 2$

Substituting $p = 2$ into the equation:

$(p + q)^2 = (q - p)^3$

$(2 + q)^2 = (q - 2)^3$

Let's find a prime number $q$ such that this equation holds.

1. Try $q = 3$:

   $(2 + 3)^2 = (3 - 2)^3$

   $5^2 = 1^3$

   $25 \neq 1$

   This does not work.

2. Try $q = 5$:

   $(2 + 5)^2 = (5 - 2)^3$

   $7^2 = 3^3$

   $49 \neq 27$

   This does not work either.

3. Try $q = 7$:

   $(2 + 7)^2 = (7 - 2)^3$

   $9^2 = 5^3$

   $81 \neq 125$

   This also does not work.

4. Try $q = 5$:

   $(2 + 5)^2 = (5 - 2)^3$

   $7^2 = 3^3$

   $49 \neq 27$

   This does not work either.

Step 3: Checking if there are other valid cases

None of these small values satisfy the equation. Hence, we try the possibility of checking where the expression is simplified to keep only positive values for both sides. Since for all $p$ and $q$ chosen above do not hold, we need to analyze for what choice $p$ and $q$ hold, ensuring both sides of the expression equate.

Step 4: Letting $q = 2p$

This assumption helps maintain positivity and equalization on both sides. We test by substituting $p = 3$, keeping $q = 3 \cdot 2$ and checking with large numbers. If it fails, we conclude that only $p = q = 2$.

Conclusion

Since only possible value is 2:

$pq = 3 \cdot 5 = 15$

Hence, the expression becomes undefined for other primes.

Thus, the correct value of $pq = 15$.

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