Find these prime numbers $p, q$?

Question

Let $p, q$ be prime numbers such that

$p = 3p_1 + 2; q = 3q_1 + 2$;

$p + q + 3$ and $3p + 3q + pq + 3$ are square numbers.

Find $p, q$?

P.S. I don't have any ideas about this problem :(

Thanks :)

Answer 1

The only solutions for $(p,q)$ are $(11,2)$ and $(2,11)$.

The equation $3p + 3q + pq + 3 = k^2$ is equivalent to $(p+3)(q+3) = k^2 + 6$. Following the observation that $(p,q) = (2,2)$ is not a solution, one of $p,q$ must be odd. It follows that $k$ is even and $k^2 + 6$ has only one factor $2$, so the other prime must be even, i.e. equal to $2$.

Without loss of generality, let $q = 2$. We have to find primes $p$ such that $p + 5 = l^2$ and $5p + 9 = k^2$. The second equation can be factored as $(k+3)(k-3) = 5p$, and it easily follows that $p = 11$ is the only solution.

Interestingly, the conditions that $p+q+3$ is a square and that $p,q$ are $2 \mod 3$ were not even needed to solve the problem.