The Proof of a Conjecture Related to Divisibility Properties of z(n)

The order of appearance of n (in the Fibonacci sequence) z(n) is defined as the smallest positive integer k for which n divides the k—the Fibonacci number Fk. Very recently, Trojovský proved that z(n) is an even number for almost all positive integers n (in the natural density sense). Moreover, he conjectured that the same is valid for the set of integers n≥1 for which the integer 4 divides z(n). In this paper, among other things, we prove that for any k≥1, the number z(n) is divisible by 2k for almost all positive integers n (in particular, we confirm Trojovský’s conjecture).

Here, we are interested in some arithmetic properties of z : Z ≥1 → Z ≥1 . For that, for an integer m ≥ 2, we denote E (m) z as the set of all n ∈ Z ≥1 for which z(n) is a multiple of m (i.e., E (m) z Therefore, the aim of this paper is to study this conjecture from a more general viewpoint. We start by providing an infinite family of prime numbers (lying in an arithmetic progression) belonging to some desired sets. More precisely, we prove the following: In particular, if k ≡ 2 or 3 (mod 4), then all prime numbers p

Remark 1.
We remark that if 4 | k, then gcd(2 k − 1, 5) = 5 (actually 4 is the order of 2 modulo 5) and so no numbers of the form 2 k − 1 + 2 k · 5 can be a prime number (for k ≥ 2). Moreover, the condition k ≡ 0 (mod 4) ensures, by the Dirichlet's theorem on arithmetic progressions, the existence of infinitely many primes p ≡ 2 k − 1 (mod 2 k · 5). Now, let us observe the following table 1: nondecreases as a function of x. Therefore, a natural question arises: Clearly, Theorem 1 of [13] solves the case k = 1, while Conjecture 1 asks about the case k = 2.
The next result shows that the answer for Question 1 is yes (in particular, it solves Conjecture 1). More precisely, we have the following: Theorem 2. Let k ≥ 2 be an integer. Then there exists a positive effective computable constant c such that for all x > e 125·8 k+5 . In particular, the natural density of E (2 k ) z is equal to 1 for all k ≥ 2.
The proof of both theorems combines Diophantine properties of z(n) with analytical tools concerning primes in arithmetic progressions.

Auxiliary Results
In this section, we present some results which will be essential tools in the proof. The first ingredient is related to the value of z(p k ) for a prime number p and k ≥ 1: [14]). We have that z(2 k ) = 3 · 2 k−1 for all k ≥ 2, and z(3 k ) = 4 · 3 k−1 for all k ≥ 1. In general, it holds that The next lemma provides the largest arithmetic progression, which contains infinitely many prime numbers, belonging completely to E (2) z .
Another well-known arithmetic function related to Fibonacci numbers is the Pisano period π : Z ≥1 → Z ≥1 for which π(n) is the smallest period of (F k (mod n)) k . The first few values of π(n) (for n ∈ [1, 20]) are (see sequence A001175 in OEIS): Observe that π(n) and z(n) have similar definitions (these functions are strongly connected as can be seen in Lemma 4). However, they have a very distinct behavior related to their parity. Indeed, π(n) is even for all n ≥ 3, while Z ≥1 \E (2) z is an infinite set (since z(5 k ) = 5 k is an odd number for all k ≥ 0).
The next result provides some divisibility properties of the Pisano period for prime numbers.
The next tool is a kind of "formula" for z(n) depending on z(p a ) for all primes p dividing n. The proof of this fact can be found in [16].
In order to prove Theorem 2, we need an analytic tool related to the profusion of integers having factorization allowing only some classes of primes. The following notation will be used throughout this work: Let P be the set of prime numbers and for an integer q ≥ 2, set P(a, q) as the set of all prime numbers of the form a + kq for some integer k ≥ 0 (Dirichlet's theorem on arithmetic progressions ensures that P(a, q) is an infinite set whenever gcd(a, q) = 1). Moreover, let B be the union of B distinct reduced residue classes modulo q. Let N B = {n ≥ 1 : p | n ⇒ p ∈ B} be the set of all positive integers whose prime factors belong exclusively to B. Additionally, denote β := B/φ(q) (where φ(n) is the Euler totient function) and which has an analytic continuation to a neighborhood of s = 1. Here, as usual, ζ(s) denotes the Riemann zeta function.
Our next auxiliary lemma is related to a work due to Chang and Martin [17]. More precisely, Lemma 6 (Theorem 3.4 of [17]). For any integer q ≥ 3, there exists a positive absolute constant C such that uniformly for q ≤ (log x) 1/3 , we have where, as usual, Γ(z) = ∞ 0 t z−1 e −t dt denotes the Gamma function.
Now, we are ready to deal with the proof of the theorems.
Since z(p) is even and s/r is odd (because so is s), the possibility z(p) = s/r is ruled out. Therefore z(p) = 2s/r ≡ 2 (mod 4).
The case k ≡ 2 or 3 (mod 4). If k = 4 + 2, then In addition, in the case k = 4 + 3, we have In any case, we can use Lemma 3 (ii) to deduce that π(p) = 2(p + 1)/t for some odd integer t. Again, we use that z(p) is even (because of p ≡ 3 (mod 4) and Lemma 2) to apply Lemma 4. Then, we obtain that That is, for some i ∈ {−1, 0}. On the other hand, p ≡ 2 k − 1 (mod 2 k · 5) and so p + 1 is a multiple of 2 k , say p + 1 = 2 k r for some integer r. Thus where we used that i + 1 ≥ 0 and t ≡ 1 (mod 2). The proof is complete.
We also obtain that and we obtain that the natural density of E (2 k ) z is equal to 1. The proof is then complete.

Further Comments
We close this paper by making some comments about the two other questions which were raised in [13], namely, Question 2. Are there infinitely many prime numbers p for which δ(E z . This does not seem to be an easy task, since it depends on a better knowledge of z(p) for prime numbers p. However, we even do not know if z(p) = p + 1 has infinitely many prime solutions. For this reason, Question 2 remains as an open problem.
On the other hand, Question 3 is too general (since nothing is required about this lower bound-we are assuming that it should be a nondecreasing function of x). In this case, we are able to answer this question reasonably as follows.