Find the number of ordered pairs of distinct positive primes p, q (p≠q) such that p^2+7pq+q^2 is the square of an integer.
To find the number of ordered pairs of distinct positive primes p and q such that p^2 + 7pq + q^2 is the square of an integer, we can use a combination of mathematical analysis and trial and error.
Let's consider the given expression: p^2 + 7pq + q^2.
We can rewrite it as (p + q)^2 + 5pq.
Now, let's assume there exists an integer k such that (p + q)^2 + 5pq = k^2.
Rearranging the equation, we get (p + q)^2 = k^2 - 5pq.
We can try different values of p and q to satisfy this equation. However, there is a restriction that p and q should be distinct positive primes.
Here's an approach to simplify the problem:
1. Start by assuming p = 2.
- Substitute p = 2 into the rearranged equation: (2 + q)^2 = k^2 - 10q.
- Rearrange and simplify: q^2 + (4 - 10k)q + (4 - k^2) = 0.
- Apply the quadratic formula: q = [(10k - 4) ± √((4 - 10k)^2 - 4(4 - k^2))] / 2.
- Simplify further and check for prime numbers.
2. Repeat the same process for different values of p (excluding 2) to find possible values of q.
3. After obtaining the sets of p and q, check if p + q is a perfect square.
4. Count the number of valid ordered pairs of distinct positive primes p and q.
Remember to check for distinct positive prime numbers p and q in each step and to verify if p + q is a perfect square.