- #1
Doom of Doom
- 86
- 0
How can I prove this?
Suppose [tex]X[/tex] is a set of 16 distinct positive integers, [tex]X=\left\{{x_{1}, \cdots , x_{16}}\right\}[/tex].
Then, for every [tex]X[/tex], there exists some integer [tex]k\in\left\{{1, \cdots , 8}\right\}[/tex] and disjoint subsets [tex]A,B\subset X[/tex]
[tex]A=\left\{a_{1},\cdots\ ,a_{k}\right\}[/tex] and [tex]B=\left\{b_{1},\cdots\ ,b_{k}\right\}[/tex]
such that [tex]\left|\alpha - \beta\right|<.00025[/tex],
where [tex]\alpha= \frac{1}{a_{1}}+\cdots+\frac{1}{a_{k}}[/tex] and [tex]\beta= \frac{1}{b_{1}}+\cdots+\frac{1}{b_{k}}[/tex].
I know that .00025 is pretty close to 2^-12.
Suppose [tex]X[/tex] is a set of 16 distinct positive integers, [tex]X=\left\{{x_{1}, \cdots , x_{16}}\right\}[/tex].
Then, for every [tex]X[/tex], there exists some integer [tex]k\in\left\{{1, \cdots , 8}\right\}[/tex] and disjoint subsets [tex]A,B\subset X[/tex]
[tex]A=\left\{a_{1},\cdots\ ,a_{k}\right\}[/tex] and [tex]B=\left\{b_{1},\cdots\ ,b_{k}\right\}[/tex]
such that [tex]\left|\alpha - \beta\right|<.00025[/tex],
where [tex]\alpha= \frac{1}{a_{1}}+\cdots+\frac{1}{a_{k}}[/tex] and [tex]\beta= \frac{1}{b_{1}}+\cdots+\frac{1}{b_{k}}[/tex].
I know that .00025 is pretty close to 2^-12.