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.

Some problems related to the growth of $z(n)$

“…In this direction, we define, for any positive integer n, the order of apparition (or the rank of appearance) of n in the Fibonacci sequence, denoted by z(n), as the minimum element of the set {k ≥ 1 : n | F k }. This function is well defined by a result of Lucas [9, p. 300] (in 1878), and in fact a simple combinatorial argument yields z(n) ≤ n 2 for all positive integers n. We note that there is not a general closed formula for the z(n), and therefore Diophantine equations related to z(n) play an important role in its best comprehension (see [10,16,17,19]). A number of authors have considered, in varying degrees of generality, the problem of determining a special closed formula for z(n), when n is a number which is related to a sum or a product of terms of Fibonacci and Lucas sequences (see, for example, [5,6,11,18] and the references therein).…”

On the Parity of the Order of Appearance in the Fibonacci Sequence

“…The arithmetic function z : Z ≥1 → Z ≥1 is well defined (as can be seen in Lucas [1], p. 300) and, in fact, z(n) ≤ 2n is the sharpest upper bound (as can be seen in [2]). A few values of z(n) (for n ∈ [1,50]) can be found in Table 1 (see the OEIS [3] sequence A001177 and for more facts on properties of z(n) see, e.g., [4][5][6][7][8][9]). Since F i = i, for i ∈ {1, 2}, the Pisano period may be defined as π(n) := min{k ≥ 1 :…”

The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of z(n)

“…Here, we are interested in some Diophantine properties of z(n) (this topic has been dealt with by many authors; see, for instance, [6][7][8][9][10][11][12]). For any integer m ≥ 2, we denote E (m) z as the set of all n ∈ Z ≥1 for which m divides z(n), that is,…”