Mathematics – Combinatorics
Scientific paper
1997-04-15
Mathematics
Combinatorics
Scientific paper
A partition of the positive integers into sets $A$ and $B$ {\em avoids} a set $S\subset\N$ if no two distinct elements in the same part have a sum in $S$. If the partition is unique, $S$ is {\em uniquely avoidable.} For any irrational $\alpha>1$, Chow and Long constructed a partition which avoids the numerators of all convergents to $\alpha$, and conjectured that the set $S_\alpha$ which this partition avoided was uniquely avoidable. We prove that the set of numerators of convergents is uniquely avoidable if and only if the continued fraction for $\alpha$ has infinitely many partial quotients equal to 1. We also construct the set $S_\alpha$ and show that it is always uniquely avoidable.
No associations
LandOfFree
Continued Fractions and Unique Additive Partitions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Continued Fractions and Unique Additive Partitions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Continued Fractions and Unique Additive Partitions will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-489914