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.