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LetF nbe the nth Fibonacci number. Order of appearancez ( n )of a natural number n is defined as smallest natural number k, such that n dividesF k    . In 1930, Lehmer proved that all solutions of equationz ( n ) = n ± 1are prime numbers. In this paper, we solve equationz ( n ) = n + ℓfor| ℓ | ∈ { 1 , … , 9 }    . Our method is based on the p-adic valuation of Fibonacci numbers.

On Diophantine Equations Related to Order of Appearance in Fibonacci Sequence