Question 36
Let $p_1$ and $q_1$ be integers such that the equation $x^2 + p_1x + q_1 = 0$ has two integer solutions. For all $n \in \mathbb{N}$ we define the numbers $p_{n+1}$ and $q_{n+1}$ by the formulas
$$
p_{n+1} = p_n + 1, \quad q_{n+1} = q_n + \frac{1}{2}p_n.
$$
Prove that there exists an infinite number of positive integers $n$ such that the equation $x^2 + p_nx + q_n = 0$ has two integer solutions.
$$
p_{n+1} = p_n + 1, \quad q_{n+1} = q_n + \frac{1}{2}p_n.
$$
Prove that there exists an infinite number of positive integers $n$ such that the equation $x^2 + p_nx + q_n = 0$ has two integer solutions.