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

Abstract

The Fibonacci sequence, whose first terms are f0; 1; 1; 2; 3; 5; : : :g, is generated using the recursive formula Fn+2 = Fn+1 + Fn with F0 = 0 and F1 = 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 f2; 1; 3; 4; 7; 11; : : :g, is generated using the recursive formula Ln+2 = Ln+1 + Ln with L0 = 2 and L1 = 1. In this paper, patterns in the prime factors of sums of powers of Fibonacci and Lucas numbers are examined. For example, F2 3n+4 + F2 3n+2 is even for all n 2 N0. To prove these results, techniques from modular arithmetic and facts about the divisibility of Fibonacci and Lucas numbers are utilized. KEYWORDS: Fibonacci Sequence; Lucas Sequence; Modular Arithmetic; Divisibility Sequence