Find all primes p and q such that p^2 + 7pq + q^2 is the square of an integer?

The answer is not important. What is the approach to this problem?

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- Jul 1st 2008, 01:05 AMfardeen_genChallenging question?
Find all primes p and q such that p^2 + 7pq + q^2 is the square of an integer?

The answer is not important. What is the approach to this problem?

- Jul 1st 2008, 03:34 AMIsomorphism
First of all, with , so any prime would do.

So from now on lets assume

Let

Now since are all primes, the factors of are

Observing that , we have

1)

Assume a solution exists. Now since q is a prime, , which is not possible since we assumed p to be prime and hence p > 1. Thus no solution exists.

2)

We have already solved this case.

3)

Same as (2)

4)

Since q is a natural number, (p-2)|9 and thus p-2 can be either 1,3 or 9.

So trying all cases,

1)

Thats (Rock)

2)

Its right but we already have tackled case.

3)

Thats (Rock) again.

So the solutions are the following(note that w represents any arbitrary prime):