What are the last three digits of the largest perfect square that can be expressed as p^2+pq+q^2, where p and q are (positive) prime numbers?
To find the last three digits of the largest perfect square that can be expressed as p^2 + pq + q^2, where p and q are positive prime numbers, we can use a simple approach.
First, let's analyze the expression p^2 + pq + q^2. Notice that this is a quadratic expression in terms of p, with q considered as a constant. The quadratic equation can be written as p^2 + (q)p + q^2.
The last three digits of a number are determined by its remainder when divided by 1000. Hence, our task is to find the remainder of p^2 + pq + q^2 when divided by 1000.
To proceed, we need to consider the possible values of q. Since p and q are prime numbers, q can only be 2 or an odd prime number. Let's analyze both cases:
Case 1: q = 2
If q = 2, the expression becomes p^2 + 2p + 4. We can try different values of p starting from 2 and calculate the remainder when dividing the expression by 1000 until we find a perfect square.
Case 2: q is an odd prime number
If q is an odd prime number, we need to consider the last three digits of q^2, as it will be added to p^2 + pq.
By testing different values of p and q, we can calculate the last three digits of the expression p^2 + pq + q^2 until we find a perfect square.
However, since you specifically asked for the largest perfect square, we can narrow down our search. We know that the largest possible value for p is a prime number less than 1000. Additionally, q^2 will be the smallest possible value of q^2 with a three-digit number.
By combining these constraints, we can iterate for p starting from the largest prime number less than 1000 and q starting from the smallest prime number greater than 30 (sqrt(1000)) until we find a perfect square.
This approach ensures that we check all possible combinations while minimizing the number of iterations required.
Please note that since the process involves iteration and calculation, it may take some time to find the answer.