Prime Factors and Divisibility of Sums of Powers of Fibonacci and Lucas Numbers.

Authors :: Karcher, Spirit
Michael, Mariah
Source :: American Journal of Undergraduate Research; Mar2021, Vol. 17 Issue 4, p59-69, 11p
Publication Year :: 2021
Abstract: The Fibonacci sequence, whose first terms are {0, 1, 1, 2, 3, 5, . . .}, is generated using the recursive formula F<subscript>n+2</subscript> = F<subscript>n+1</subscript> + F<subscript>n</subscript> with F<subscript>0</subscript> = 0 and F<subscript>1</subscript> = 1. This sequence is one of the most famous integer sequences because of its fascinating mathematical properties and connections with other fields such as biology, art, and music. Closely related to the Fibonacci sequence is the Lucas sequence. The Lucas sequence, whose first terms are {2, 1, 3, 4, 7, 11, . . .}, is generated using the recursive formula L<subscript>n+2</subscript> = L<subscript>n+1</subscript> + L<subscript>n</subscript> with L<subscript>0</subscript> = 2 and L<subscript>1</subscript> = 1. In this paper, patterns in the prime factors of sums of powers of Fibonacci and Lucas numbers are examined. For example, F² <subscript>3n+4</subscript> + F² <subscript>3n+2</subscript> is even for all n - N<subscript>0</subscript>. To prove these results, techniques from modular arithmetic and facts about the divisibility of Fibonacci and Lucas numbers are utilized. [ABSTRACT FROM AUTHOR]

Subjects :: LUCAS numbers
MODULAR arithmetic
FIBONACCI sequence
DIVISIBILITY groups
INTEGERS

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