On the greatest common divisor of $n$ and the $n$th Fibonacci number

TL;DR

This paper investigates the set of all gcds of positive integers and their corresponding Fibonacci numbers, establishing their density properties and providing explicit formulas for related prime distributions.

Contribution

It proves that the set of gcds has zero asymptotic density and grows at least as fast as x / log x, using recent results on prime divisibility in Fibonacci sequences.

Findings

The set of gcds has zero asymptotic density.

The number of such gcds up to x is at least proportional to x / log x.

Explicit formulas for prime divisibility related to Fibonacci numbers are utilized.

Abstract

Let $\mathcal{A}$ be the set of all integers of the form $\gcd(n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$th Fibonacci number. We prove that $\#\left(\mathcal{A} \cap [1, x]\right) \gg x / \log x$ for all $x \geq 2$, and that $\mathcal{A}$ has zero asymptotic density. Our proofs rely on a recent result of Cubre and Rouse which gives, for each positive integer $n$, an explicit formula for the density of primes $p$ such that $n$ divides the rank of appearance of $p$, that is, the smallest positive integer $k$ such that $p$ divides $F_k$.

Equations

$$\#\mathcal{B}(x)\le x^{1-\left(\frac{1}{2}+o(1)\right)\log \log \log x/\log \log x},$$

$$1,2,5,7,10,12,13,17,24,25,26,29,34,35,36,37,\dots$$

$$\zeta (m):=\underset{x\to +\infty }{\lim }\frac{\#\{p\le x:m\mid z(p)\}}{x/\log x}.$$

$$\zeta (q^e)=\frac{q^{2-e}}{q^2-1},$$

$$\zeta (m)=\prod _{q^e\mid \mid m}\zeta (q^e)\cdot \{\begin{array}{ll}1& \text{ if }10\nmid m,\\ \frac{5}{4}& \text{ if }m\equiv 10\mathrm{mod}20,\\ \frac{1}{2}& \text{ if }20\mid m.\end{array}$$

$$\sum _{q>y}\frac{1}{\phi (\ell (q))}\ll \frac{1}{y^{1/4}},$$

$$2^{\#{\mathcal{Q}}_{\gamma }(x)}\le \prod _{p\in {\mathcal{Q}}_{\gamma }(x)}p\mid \prod _{n\le x^{\gamma }}{F}_n\le 2^{{\sum }_{n\le x^{\gamma }}n}\le 2^{O(x^{2\gamma })},$$

$$\sum _{q>y}\frac{1}{\phi (\ell (q))}\ll \sum _{q>y}\frac{\log \log \ell (q)}{\ell (q)}\ll \sum _{q>y}\frac{\log \log q}{\ell (q)}\ll \sum _{q>y}\frac{q^{\epsilon }}{\ell (q)},$$

$$\sum _{\begin{array}{c}q>y\\ q\notin {\mathcal{Q}}_{\gamma }\end{array}}\frac{q^{\epsilon }}{\ell (q)}\ll \sum _{\begin{array}{c}q>y\\ q\notin {\mathcal{Q}}_{\gamma }\end{array}}\frac{1}{q^{1-\epsilon }z(q)}\le \sum _{q>y}\frac{1}{q^{1+\gamma -\epsilon }}\ll {\int }_y^{+\infty }\frac{\mathrm{d}t}{t^{1+\gamma -\epsilon }}\ll \frac{1}{y^{\gamma -\epsilon }}.$$

$$\sum _{\begin{array}{c}q>y\\ q\in {\mathcal{Q}}_{\gamma }\end{array}}\frac{q^{\epsilon }}{\ell (q)}$$

$$\le {\int }_y^{+\infty }\frac{\#{\mathcal{Q}}_{\gamma }(t)}{t^{2-\epsilon }}\mathrm{d}t\ll {\int }_y^{+\infty }\frac{\mathrm{d}t}{t^{2-2\gamma -\epsilon }}\ll \frac{1}{y^{1-2\gamma -\epsilon }}.$$

$$\pi (x,m,a):=\#\{p\le x:p\equiv a\mathrm{mod}m\}.$$

$$\pi (x,m,a)<\frac{2x}{\phi (m)\log (x/m)},$$

$${\mathcal{P}}_1$$

$${\mathcal{P}}_2$$

$$\mathcal{P}$$

$$\underset{x\to +\infty }{\lim }\frac{\#{\mathcal{P}}_1(x)}{x/\log x}$$

$$=\prod _{3\le q\le y}\left(1-\zeta (q)\right)=\prod _{3\le q\le y}\left(1-\frac{q}{q^2-1}\right),$$

$$\#{\mathcal{P}}_1(x)\ge \frac{1}{2}\prod _{3\le q\le y}\left(1-\frac{q}{q^2-1}\right)\cdot \frac{x}{\log x}\gg \frac{1}{\log y}\cdot \frac{x}{\log x},$$

$$\#{\mathcal{P}}_2(x)\le \sum _{q>y}\#\{p\le x:\ell (q)\mid z(p)\}\le \sum _{q>y}\pi (x,\ell (q),\pm 1),$$

$$\pi (x,\ell (q),\pm 1):=\pi (x,\ell (q),-1)+\pi (x,\ell (q),1).$$

$$\sum _{y<q<x^{1/2}}\pi (x,\ell (q),\pm 1)\ll \sum _{q>y}\frac{1}{\phi (\ell (q))}\cdot \frac{x}{\log x}\ll \frac{1}{y^{1/4}}\cdot \frac{x}{\log x}.$$

$$\sum _{q>x^{1/2}}\pi (x,\ell (q),\pm 1)\ll \sum _{q>x^{1/2}}\frac{x}{\ell (q)}\le \sum _{q>x^{1/2}}\frac{x}{\phi (\ell (q))}\ll x^{7/8}.$$

$$\#{\mathcal{P}}_2(x)\ll \frac{1}{y^{1/4}}\cdot \frac{x}{\log x}+x^{7/8}.$$

$$\#\mathcal{A}(x)\gg \#\mathcal{P}(x)\ge \#{\mathcal{P}}_1(x)-\#{\mathcal{P}}_2(x)\ge \left(\frac{c_1}{\log y}-\frac{c_2}{y^{1/4}}-\frac{c_2\log x}{x^{1/8}}\right)\cdot \frac{x}{\log x},$$

$$\frac{c_1}{\log y}-\frac{c_2}{y^{1/4}}>0.$$

$$\prod _{p\in \mathcal{P}(y)}\left(1-\frac{1}{p}\right)<\epsilon .$$

$$\underset{x\to +\infty }{\mathrm{limsup}}\frac{\#\mathcal{A}(x)}{x}\le \underset{x\to +\infty }{\mathrm{limsup}}\frac{\#{\mathcal{A}}_1(x)}{x}+\underset{x\to +\infty }{\mathrm{limsup}}\frac{\#{\mathcal{A}}_2(x)}{x}\le \prod _{p\in \mathcal{P}(y)}\left(1-\frac{1}{p}\right)+\frac{1}{q}<2\epsilon ,$$

Full text

1. Introduction

Let $(F_{n})_{n\geq 1}$ be the sequence of Fibonacci numbers, defined as usual by $F_{1}=F_{2}=1$ and $F_{n+2}=F_{n+1}+F_{n}$, for all positive integers $n$. Moreover, let $g$ be the arithmetic function defined by $g(n):=\gcd(n,F_{n})$, for each positive integer $n$. The first values of $g$ are listed in OEIS A104714 [13].

The set $\mathcal{B}$ of fixed points of $g$, i.e., the set of positive integers $n$ such that $n$ divides $F_{n}$, has been studied by several authors. For instance, André-Jeannin [2] and Somer [14] investigated the arithmetic properties of the elements of $\mathcal{B}$. Furthermore, Luca and Tron [8] proved that

[TABLE]

when $x\to+\infty$, and Sanna [12] generalized their result to Lucas sequences. More generally, the study of the distribution of positive integers $n$ dividing the $n$th term of a linear recurrence has been studied by Alba González, Luca, Pomerance, and Shparlinski [1], while, Corvaja and Zannier [4], and Sanna [10] considered the distribution of positive integers $n$ such that the $n$th term of a linear recurrence divides the $n$th term of another linear recurrence. Also, it follows from a result of Sanna [11] that the set $g^{-1}(1)$, i.e., the set of positive integers $n$ such that $n$ and $F_{n}$ are relatively prime, has a positive asymptotic density.

Define $\mathcal{A}:=\{g(n):n\geq 1\}$. Note that, in particular, $\mathcal{B}\subseteq\mathcal{A}$. The aim of this article is to study the structural properties and the distribution of the elements of $\mathcal{A}$. Note that it is not immediately clear whether or not a given positive integer belongs to $\mathcal{A}$. To this aim, we provide in §2 an effective criterion which allows us to enumerate the elements of $\mathcal{A}$, in increasing order, as:

[TABLE]

Our first result is a lower bound for the counting function of $\mathcal{A}$.

Theorem 1.1**.**

$\#\mathcal{A}(x)\gg x/\log x$, for all $x\geq 2$.

It is worth noting that it follows at once from Theorem 1.1 and (1) that $\mathcal{B}$ has zero asymptotic density relative to $\mathcal{A}$ (we omit the details):

Corollary 1.2**.**

$\#\mathcal{B}(x)=o(\#\mathcal{A}(x))$, as $x\to+\infty$.

Our second result is that $\mathcal{A}$ has zero asymptotic density:

Theorem 1.3**.**

$\#\mathcal{A}(x)=o(x)$, as $x\to+\infty$.

It would be nice to have an effective upper bound for $\#\mathcal{A}(x)$ or, even better, to obtain its asymptotic order of growth. We leave these as open questions for the interested readers.

Notation

Throughout, we reserve the letters $p$ and $q$ for prime numbers. Moreover, given a set $\mathcal{S}$ of positive integers, we define $\mathcal{S}(x):=\mathcal{S}\cap[1,x]$ for all $x\geq 1$. We employ the Landau–Bachmann “Big Oh” and “little oh” notations $O$ and $o$, as well as the associated Vinogradov symbols $\ll$ and $\gg$. In particular, all the implied constants are intended to be absolute, unless it is explicitly stated otherwise.

2. Preliminaries

This section is devoted to some preliminary results needed in the later proofs. For each positive integer $n$, let $z(n)$ be rank of appearance of $n$ in the sequence of Fibonacci numbers, that is, $z(n)$ is the smallest positive integer $k$ such that $n$ divides $F_{k}$. It is well known that $z(n)$ exists. All the statements in the next lemma are well known, and we will use them implicitly without further mention.

Lemma 2.1**.**

For all positive integer $m,n$ and all prime numbers $p$, we have:

1. (i)

$F_{m}\mid F_{n}$* whenever $m\mid n$.* 2. (ii)

$m\mid F_{n}$* if and only if $z(m)\mid n$.* 3. (iii)

$z(m)\mid z(n)$* whenever $m\mid n$.* 4. (iv)

$z(p)\mid p-\left(\frac{p}{5}\right)$, where $\left(\frac{p}{5}\right)$ is a Legendre symbol.

For each positive integer $n$, define $\ell(n):=\operatorname{lcm}(n,z(n))$. The next lemma shows some elementary properties of the functions $g$, $\ell$, $z$, and their relationship with $\mathcal{A}$.

Lemma 2.2**.**

For all positive integer $m,n$ and all prime numbers $p$, we have:

1. (i)

$g(m)\mid g(n)$* whenever $m\mid n$.* 2. (ii)

$n\mid g(m)$* if and only if $\ell(n)\mid m$.* 3. (iii)

$n\in\mathcal{A}$* if and only if $n=g(\ell(n))$.* 4. (iv)

$p\mid n$* whenever $\ell(p)\mid\ell(n)$ and $n\in\mathcal{A}$.* 5. (v)

$\ell(p)=pz(p)$* whenever $p\neq 5$, and $\ell(5)=5$.* 6. (vi)

$p\in\mathcal{A}$* if $p\neq 3$ and $\ell(q)\nmid z(p)$ for all prime numbers $q$.*

Proof.

Facts (i) and (ii) follow easily from the definitions of $g$ and $\ell$ and the properties of $z$. To prove (iii), note that $n$ divides both $\ell(n)$ and $F_{\ell(n)}$ hence $n\mid g(\ell(n))$ for all positive integers $n$. Conversely, if $n\in\mathcal{A}$, then $n=g(m)$ for some positive integer $m$. In particular, $n\mid g(m)$ which is equivalent to $\ell(n)\mid m$ by (ii). Therefore $g(\ell(n))\mid g(m)=n$, thanks to (i), and in conclusion $g(\ell(n))=n$. Fact (iv) follows at once from (ii) and (iii).

A quick computation shows that $\ell(5)=5$, while for all prime numbers $p\neq 5$ we have $\gcd(p,z(p))=1$, since $z(p)\mid p\pm 1$, so that $\ell(p)=pz(p)$, and this proves (v).

Lastly, let us suppose that $p\neq 3$ is a prime number such that $\ell(q)\nmid z(p)$ for all prime numbers $q$. In particular, $p\neq 5$ since $\ell(5)=z(5)=5$, by (v). Also, the claim (vi) is easily seen to hold for $p=2$. Hence, let us suppose hereafter that $p\geq 7$. Since $z(p)\mid p\pm 1$, it easily follows that $p\mid\mid g(\ell(p))$. At this point, if $q\mid g(\ell(p))$ for some prime $q\neq p$, then $\ell(q)\mid\ell(p)=pz(p)$ thanks to (ii). But $\ell(q)\nmid z(p)$, hence $p\mid\ell(q)=\operatorname{lcm}(q,z(q))$ so that $p\mid z(q)\leq q+1$. Similarly, $q\mid g(\ell(p))\mid\ell(p)$ implies $q\mid z(p)\leq p+1$. Hence $|p-q|\leq 1$, which is impossible since $p\geq 7$. Therefore $q\nmid g(\ell(p))$, with the consequence that $p=g(\ell(p))$, i.e., $p\in\mathcal{A}$ by (iii). This concludes the proof of (vi). ∎

It is worth noting that Lemma 2.2(iii) provides an effective criterion to establish whether a given positive integer belongs to $\mathcal{A}$ or not. This is how we evaluated the elements of $\mathcal{A}$ listed in the introduction.

It follows from a result of Lagarias [6, 7], that the set of prime numbers $p$ such that $z(p)$ is even has a relative density of $2/3$ in the set of all prime numbers. Bruckman and Anderson [3, Conjecture 3.1] conjectured, for each positive integer $m$, a formula for the limit

[TABLE]

Their conjecture was proved by Cubre and Rouse [5, Theorem 2], who obtained the following result.

Theorem 2.3**.**

For each prime number $q$ and each positive integer $e$, we have

[TABLE]

while for any positive integer $m$, we have

[TABLE]

Note that the arithmetic function $\zeta$ is not multiplicative. However, the restriction of $\zeta$ to the odd positive integers is multiplicative. This fact will be useful later.

Let $\varphi$ be the Euler’s totient function. We need the following technical lemma.

Lemma 2.4**.**

We have

[TABLE]

for all $y>0$.

Proof.

For $\gamma>0$, put $\mathcal{Q}_{\gamma}:=\{p:z(p)<p^{\gamma}\}$. Clearly,

[TABLE]

from which it follows that $\mathcal{Q}_{\gamma}(x)\ll x^{2\gamma}$.

Fix also $\varepsilon\in{]0,1-2\gamma[}$. For the rest of this proof, all the implied constants may depend on $\gamma$ and $\varepsilon$. Since $\varphi(n)\gg n/\log\log n$ for all positive integers $n$ [15, Ch. I.5, Theorem 4], while, by Lemma 2.2(v), $\ell(q)\ll q^{2}$ for all prime numbers $q$, we have

[TABLE]

for all $y>0$.

On the one hand, again by Lemma 2.2(v),

[TABLE]

On the other hand, by partial summation,

[TABLE]

The claim follows by putting together (2), (3), and (2), and by choosing $\gamma=1/3$ and $\varepsilon=1/12$. ∎

Lastly, for all relatively prime integers $a$ and $m$, define

[TABLE]

We need the following version of the Brun–Titchmarsh theorem [9, Theorem 2].

Theorem 2.5**.**

If $a$ and $m$ are relatively prime integers and $m>0$, then

[TABLE]

for all $x>m$.

3. Proof of Theorem 1.1

First, since $1\in\mathcal{A}$, it is enough to prove the claim only for all sufficiently large $x$. Let $y>5$ be a real number to be chosen later. Define the following sets of primes:

[TABLE]

We have $\mathcal{P}\subseteq\mathcal{A}\cup\{3\}$. Indeed, since $3\mid\ell(2)$ and $q\mid\ell(q)$ for each prime number $q$, it follows easily that if $p\in\mathcal{P}$ then $\ell(q)\nmid z(p)$ for all prime numbers $q$, which, by Lemma 2.2(vi), implies that $p\in\mathcal{A}$ or $p=3$.

Now we give a lower bound for $\#\mathcal{P}_{1}(x)$. Let $P_{y}$ be the product of all prime numbers in $[3,y]$, and let $\mu$ be the Möbius function. By using the inclusion-exclusion principle and Theorem 2.3, we get that

[TABLE]

where we also made use of the fact that the restriction of $\zeta$ to the odd positive integers is multiplicative.

As a consequence, for all sufficiently large $x$ depending only on $y$, let say $x\geq x_{0}(y)$, we have

[TABLE]

where the last inequality follows from Mertens’ third theorem [15, Ch. I.1, Theorem 11].

We also need an upper bound for $\#\mathcal{P}_{2}(x)$. Since $z(p)\mid p\pm 1$ for all primes $p>5$, we have

[TABLE]

for all $x>0$, where, for the sake of brevity, we put

[TABLE]

On the one hand, by Theorem 2.5 and Lemma 2.4, we have

[TABLE]

On the other hand, by the trivial estimate for $\pi(x,\ell(q),\pm 1)$ and Lemma 2.4, we get

[TABLE]

Therefore, putting together (5), (6), and (7), we find that

[TABLE]

In conclusion, there exist two absolute constants $c_{1},c_{2}>0$ such that

[TABLE]

for all $x\geq x_{0}(y)$.

Finally, we can choose $y$ to be sufficiently large so that

[TABLE]

Hence, from (8) it follows that $\#\mathcal{A}(x)\gg x/\log x$, for all sufficiently large $x$.

4. Proof of Theorem 1.3

Fix $\varepsilon>0$ and pick a prime number $q$ such that $1/q<\varepsilon$. Let $\mathcal{P}$ be the set of prime numbers $p$ such that $\ell(q)\mid z(p)$. By Theorem 2.3, we know that $\mathcal{P}$ has a positive relative density in the set of primes. As a consequence, we can pick a sufficiently large $y>0$ so that

[TABLE]

Let $\mathcal{B}$ be the set of positive integers without prime factors in $\mathcal{P}(y)$. We split $\mathcal{A}$ into two subsets: $\mathcal{A}_{1}:=\mathcal{A}\cap\mathcal{B}$ and $\mathcal{A}_{2}:=\mathcal{A}\setminus\mathcal{A}_{1}$. If $n\in\mathcal{A}_{2}$ then $n$ has a prime factor $p$ such that $\ell(q)\mid z(p)$. Hence, $\ell(q)\mid\ell(n)$ and, by Lemma 2.2(iv), we get that $q\mid n$, so all the elements of $\mathcal{A}_{2}$ are multiples of $q$. In conclusion,

[TABLE]

and, by the arbitrariness of $\varepsilon$, it follows that $\mathcal{A}$ has zero asymptotic density.