Good positive integers

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Good positive integers

Unread post by tanmoy » Sun Sep 27, 2015 7:16 pm

For a positive real number $c>0$,call a positive integer $c$-$good$ if for all positive integer $m<n$,$\frac {m} {n}$ can be represented as
$\frac{m}{n}=\frac{a_{0}}{b_{0}}+...+\frac{a_{k}} {b_{k}}$
for some non-negative integers $k<\frac{n} {c}$ and $2b_{i}\leq n$ and $0\leq a_{i}<b_{j}$ and $0\leq j\leq k$.Show that,for any real $c$,there are an infinite number of $c$-$good$ positive integers.
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Re: Good positive integers

Unread post by Masum » Sat Oct 10, 2015 12:20 am

Aha. That problem of mine! It's quite easy actually. But it can be made a bit hard with more restrictions. Have you solved it?
One one thing is neutral in the universe, that is $0$.

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