OME 2020 Problem 6

Let $S$ be a finite set of integers. We define $d_2(S)$ and $d_3(S)$ as: $\bullet$ $d_2(S)$ is the number of elements $a \in S$ such that there exist $x, y \in \mathbb{Z}$ such that $x^2-y^2 = a$ $\bullet$ $d_3(S)$ is the number of elements $a \in S$ such that there exist $x, y \in \mathbb{Z}$ such that $x^3-y^3 = a$ (a) Let $m$ be an integer and $S = \{m, m+1, \ldots, m+2019\}$ . Prove: $$d_2(S) > \frac{13}{7} d_3(S)$$ (b) Let $S_n = \{1, 2, \ldots, n\}$ with $n$ a positive integer. Prove that there exists a $N$ so that for all $n > N$ : $$d_2(S_n) > 4 \cdot d_3(S_n)$$