### Faculty Member

This project continues some of the themes explored in our Fall 2017

IGL project. A Beatty sequence is a sequence of the form *[a n]*,

where *a* is an irrational number and the bracket denotes the floor

function. A remarkable result, called Beatty's Theorem, says that if

*a* and *b* are irrational numbers such that *1/a+1/b=1*, then the associated Beatty sequences "partition" the

natural numbers. That is, every natural number belongs to

exactly one of these two sequences. These partitions are related to

generalized, or "exotic", number systems such as representations of

integers as sums of Fibonacci numbers.

It is known that Beatty's Theorem does not extend directly to

partitions into three or more sets, and finding appropriate analogs

of Beatty's Theorem for such partitions is an interesting, and wide

open, problem. The goal of this project is to explore this and

related questions, both experimentally and theoretically.

### Team Meetings

### Project Difficulty

### Undergrad Prerequisites

Completion of Calculus 3. Exposure to an upper level proof-based

course such as elementary number theory, real or complex analysis

would be desirable, though specific knowledge in

any of these areas is not required. Prior experience with Mathematica

(preferred) or other appropriate software such as Python would

be helpful, but the necessary skills can also be acquired at the

beginning of the project.