Beatty Sequences, Exotic Number Systems, and Partitions of the Integers

Faculty Member

Ken Stolarsky and A J Hildebrand

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.