On the Diophantine Equation z(n) = (2 − 1/k)n Involving the Order of Appearance in the Fibonacci Sequence

Abstract

Let ( F n ) n ≥ 0 be the sequence of the Fibonacci numbers. The order (or rank) of appearance z ( n ) of a positive integer n is defined as the smallest positive integer m such that n divides F m . In 1975, Sallé proved that z ( n ) ≤ 2 n , for all positive integers n. In this paper, we shall solve the Diophantine equation z ( n ) = ( 2 − 1 / k ) n for positive integers n and k.

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 :…”

On Some Properties of the Limit Points of (z(n)/n)n

“…In 2020, Trojovská [8] solved completely the Diophantine equation z(n) = (2 − 1/k)n, with k ≥ 1. She proved that all solutions are related to the cases k = 1 or 2 (and there are two infinite families of solutions for each of these cases).…”