Question 153
For an integer $n \gt 0$, denote by $\mathcal{F}(n)$ the set of integers $m \gt 0$ for which the polynomial $p(x) = x^2 + mx + n$ has an integer root.
a. Let $S$ denote the set of integers $n \gt 0$ for which $\mathcal{F}(n)$ contains two consecutive integers. Show that $S$ is infinite but
$$
\sum_{n \in S} \frac{1}{n} \le 1.
$$
b. Prove that there are infinitely many positive integers $n$ such that $\mathcal{F}(n)$ contains three consecutive integers.
a. Let $S$ denote the set of integers $n \gt 0$ for which $\mathcal{F}(n)$ contains two consecutive integers. Show that $S$ is infinite but
$$
\sum_{n \in S} \frac{1}{n} \le 1.
$$
b. Prove that there are infinitely many positive integers $n$ such that $\mathcal{F}(n)$ contains three consecutive integers.