Question 68
Prove that there are infinitely many positive integers $m$ for which there exist consecutive odd positive integers $p_m, q_m$ ($\equiv p_m + 2$) such that the pairs $(p_m, q_m)$ are all distinct and
$$
p_m^2 + p_m q_m + q_m^2, \quad p_m^2 + m p_m q_m + q_m^2
$$
are both perfect squares.
$$
p_m^2 + p_m q_m + q_m^2, \quad p_m^2 + m p_m q_m + q_m^2
$$
are both perfect squares.