TSTST 2018 Problem 4

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