On Diophantine Equations Related to Order of Appearance in Fibonacci Sequence

Abstract

Let F n be the nth Fibonacci number. Order of appearance z ( n ) of a natural number n is defined as smallest natural number k, such that n divides F k . 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 | ℓ | ∈ { 1 , … , 9 } . Our method is based on the p-adic valuation of Fibonacci numbers.