Fibonacci and Lucas Congruences and Their Applications

Abstract

In this paper we obtain some new identities containing Fibonacci and Lucas numbers. These identities allow us to give some congruences concerning Fibonacci and Lucas numbers such as L2mn+k equivalent to (-1)((m+1)n) L-k (mod L-m), F2mn+k equivalent to (-1)((m=1)n) F-k (mod L-m), L2mn+k equivalent to (-1)(mn) L-k (mod F-m) and F2mn+k equivalent to (-1)(mn) F-k(mod F-m). By the achieved identities, divisibility properties of Fibonacci and Lucas numbers are given. Then it is proved that there is no Lucas number L-n such that L-n = L(2)k(t)L(m)x(2) for m > 1 and k >= 1. Moreover it is proved that L-n = LmLr is impossible if m and r are positive integers greater than 1. Also, a conjecture concerning with the subject is given.