Question 25
Let $S$ be a finite subset of the integers. Define:
- $d_2(S)$ as the number of $a\in S$ expressible as $x^2-y^2$ with integers $x,y$;
- $d_3(S)$ as the number of $a\in S$ expressible as $x^3-y^3$ with integers $x,y$.
a. For an integer $m$ and $S=\{m,m+1,\ldots,m+2019\}$, prove
$$d_2(S)\gt \frac{13}{7}d_3(S).$$
b. For a positive integer $n$, let $S_n=\{1,2,\ldots,n\}$. Prove that there is $N$ such that for every $n\gt N$,
$$d_2(S_n)\gt 4d_3(S_n).$$
- $d_2(S)$ as the number of $a\in S$ expressible as $x^2-y^2$ with integers $x,y$;
- $d_3(S)$ as the number of $a\in S$ expressible as $x^3-y^3$ with integers $x,y$.
a. For an integer $m$ and $S=\{m,m+1,\ldots,m+2019\}$, prove
$$d_2(S)\gt \frac{13}{7}d_3(S).$$
b. For a positive integer $n$, let $S_n=\{1,2,\ldots,n\}$. Prove that there is $N$ such that for every $n\gt N$,
$$d_2(S_n)\gt 4d_3(S_n).$$