On Diophantine Equations Related to Order of Appearance in Fibonacci Sequence

Abstract

Let Fn be the n-th Fibonacci number. Order of appearance z(n) of a natural number n is defined as smallest natural number k, such that n divides Fk. In 1930, Lehmer proved that all solutions of equation z(n)=n+/-1 are prime numbers. In this paper, we solve equation z(n)=n+l for the absolute value of l is from the set {1,…,9}. Our method is based on the p-adic valuation of Fibonacci numbers.