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?
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
2)
Its right but we already have tackled case.
3)
Thats again.
So the solutions are the following(note that w represents any arbitrary prime):