The order of appearance of the product of two Fibonacci and Lucas numbers

Abstract

Let $$F_{n}$$ and $$L_{n}$$ be the nth Fibonacci and Lucas number, respectively. The order of appearance is defined as the smallest natural number k such that n divides $$F_{k}$$ and denoted by z(n) . In this paper, we give explicit formulas for the terms $$ z(F_{a}F_{b}) $$ , $$ z( L_{a}L_{b}) $$ , $$ z(F_{a}L_{b}) $$ and $$ z(F_{n}F_{n+p}F_{n+2p}) $$ with $$p\ge 3$$ prime.