# Combinatorial Conundrums Can Conceal Continued Fractions

### Timothy Chow

### Mathematics Department

University of Michigan

### Tuesday, February 17, 1998

4:00 PM

### Room 102, Bradley Hall

Can you show that the positive integers can be uniquely divided into
two disjoint sets such that no two distinct numbers from the same set sum
to a Fibonacci number? D. Silverman posed this problem twenty years ago
and it was solved and generalized by many people, including K. Alladi,
P. Erdos, V. E. Hoggatt, Jr., and R. Evans. Interest in the problem then
lapsed, perhaps because it did not seem very deep. Then about a year ago,
Chris Long and I proved two unexpected theorems that related this problem
to continued fractions. This vastly generalized previous work and suggested
that there is much more to the problem than there appears to be at first
glance. There remain many unanswered questions, some of which are accessible
enough that they can be (and indeed have been) investigated by undergraduates.

The talk will be elementary and will not assume any knowledge of continued
fractions.