#### Event Title

Quantifying Carries When Adding Zeckendorf Representations of Natural Numbers

#### Faculty Sponsor(s)

Emma Wright

#### Location

Hartman Union Building Courtroom

#### Presentation Type

Event

#### Start Date

5-3-2018 3:00 PM

#### End Date

5-3-2018 4:00 PM

#### Abstract

From a combinatorial perspective, we can count the number of carries that are needed to perform any arithmetic computation in base-b, where b is any natural number, because the number of carries is independent of the algorithm used to perform the arithmetic. The Zeckendorf representation of the natural numbers is a representation that utilizes the Fibonacci sequence as its base. It has been discovered that we can use a similar approach to count arithmetic carries in the Zeckendorf base, but there are multiple types of carries in this base, and this approach only detects one of the two. We propose a method using techniques from linear algebra for counting carries that is able to detect both types of carries in Zeckendorf arithmetic. Further, we are able to count the minimal number of carries need to perform arithmetic computations in this base.

Quantifying Carries When Adding Zeckendorf Representations of Natural Numbers

Hartman Union Building Courtroom

From a combinatorial perspective, we can count the number of carries that are needed to perform any arithmetic computation in base-b, where b is any natural number, because the number of carries is independent of the algorithm used to perform the arithmetic. The Zeckendorf representation of the natural numbers is a representation that utilizes the Fibonacci sequence as its base. It has been discovered that we can use a similar approach to count arithmetic carries in the Zeckendorf base, but there are multiple types of carries in this base, and this approach only detects one of the two. We propose a method using techniques from linear algebra for counting carries that is able to detect both types of carries in Zeckendorf arithmetic. Further, we are able to count the minimal number of carries need to perform arithmetic computations in this base.