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?
So from now on lets assume
Now since are all primes, the factors of are
Observing that , we have
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.
We have already solved this case.
Same as (2)
Since q is a natural number, (p-2)|9 and thus p-2 can be either 1,3 or 9.
So trying all cases,
Its right but we already have tackled case.
So the solutions are the following(note that w represents any arbitrary prime):