**Undergraduate #69**

**Discipline:**Mathematics and Statistics

**Subcategory:**

**Ayana Ashae Tiller**

**- Savannah State University**

The Fibonacci numbers are sequences of numbers of the form: 0, 1, 1, 2, 3, 5, 8, 13… In mathematical terms, it is defined by the following recurrence relation: The first number of the sequence is 0, the second number is 1, and each subsequent number is equal to the sum of the previous two numbers of the sequence itself. The Lucas numbers are sequences of numbers of the form: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123… In mathematical terms, it is defined by the following mathematical recurrence relation L0=2 L1=1 L = L +L for n 1. Each subsequent number is equal to the sum of the previous two numbers of the sequence itself. Observe that the Lucas numbers have a slight variation of the Fibonacci numbers with different initial conditions and the two sequences are intimately related to each other as we will see later. The Fibonacci and Lucas sequences are the shining stars in the vast array of integer sequences. The sequences have a long history and special importance in mathematics, with many beautiful and intriguing properties and patterns. These numbers are also famous for possessing an amazing close relationship between the two. Mathematicians have been fascinated for centuries by the properties, patterns, and close relationships between Fibonacci and Lucas numbers. Among numerical sequences, these number sequences have achieved a kind of celebrity status. In this research, I examine some of the interesting patterns and close relationships between the Fibonacci and Lucas numbers. I primarily delve into the important results between the two numbers mostly using the principle of mathematical induction (my own proofs).

**Funder Acknowledgement(s):** Peach State Louis Stokes Alliance for Minority Participation

**Faculty Advisor: **Mulatu Lemma,
lemmam@savannahstate.edu

**Role:** I was responsible for understanding the properties and characteristics of each of the celebrity numbers that has already been founded and proved. I then worked closely with my research mentor, Mulatu Lemma to contribute to exploring an alternative way to distinguish the two numbers. We chose to explore the patterns between Fibonacci and Lucas numbers using mathematical induction since I am scheduled to take the Number Theory course next spring. We learned that mathematical induction served a unique purpose in the two celebrity numbers.