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, q (p≠q) such that p^2+7pq+q^2 is the square of an integer, we can follow these steps:

Step 1: Understand the problem
We are given that p^2+7pq+q^2 is the square of an integer. We need to find the number of ordered pairs of distinct positive primes p and q that satisfy this condition.

Step 2: Analyze the equation
Let's take a closer look at the expression p^2+7pq+q^2. We can factor it as follows:

p^2+7pq+q^2 = (p+q)(p+q+6q)

We know that the expression is the square of an integer. This means that (p+q) and (p+q+6q) must be perfect squares.

Step 3: Find perfect squares
Let's consider the factors (p+q) and (p+q+6q) separately. We need to find perfect squares for each of them.

For (p+q) to be a perfect square, p and q must have the same parity (odd or even), since only then their sum will be even. Let's consider two cases:

Case 1: Both p and q are odd primes
In this case, p = 2k+1 and q = 2l+1, where k and l are positive integers. Substituting these values into (p+q), we get:

(p+q) = (2k+1) + (2l+1) = 2(k+l+1)

For (p+q) to be a perfect square, k+l+1 must be a perfect square. Let's call it m^2.

(m^2)^2 = 4(k+l+1)

This means that (k+l+1) is also a perfect square. Since k and l are positive integers, this is equivalent to finding the number of positive integer solutions of the equation (k+l+1) = n^2, where n is a positive integer.

Case 2: Both p and q are even primes
In this case, p = 2k and q = 2l, where k and l are positive integers. Substituting these values into (p+q), we get:

(p+q) = (2k) + (2l) = 2(k+l)

Similarly to the previous case, for (p+q) to be a perfect square, k+l must be a perfect square. Let's call it m^2.

(m^2)^2 = 4(k+l)

This means that (k+l) is a perfect square. Again, we need to find the number of positive integer solutions of the equation (k+l) = n^2, where n is a positive integer.

Step 4: Count the solutions
In each case, we need to count the number of positive integer solutions for (k+l+1) = n^2 and (k+l) = n^2, respectively. This can be done by finding the number of perfect squares less than or equal to a certain integer. We can start with a small value of n and increment it until the square becomes greater than the given limit.

For example, let's say we want to find the number of solutions for n ≤ 100. We can iterate over all values of n from 1 to 10, and for each value of n, count the number of perfect squares less than or equal to n^2.

By summing up the number of solutions from both cases, we can determine the total number of ordered pairs (p, q) that satisfy the given condition.

Note: The above steps outline the general approach to solving the problem. The actual implementation might involve more details and optimizations to handle larger values of p and q.