On numbers $n$ relatively prime to the $n$th term of a linear recurrence

TL;DR

This paper investigates the set of positive integers n for which a linear recurrence term u_n is coprime to n, establishing the existence of an asymptotic density and conditions for its positivity.

Contribution

It proves the existence of an asymptotic density for the set of n where u_n and n are coprime, and characterizes when this density is positive.

Findings

The set of such n has an asymptotic density.

The density is positive unless (u_n / n) forms a linear recurrence.

Provides conditions under which the density is zero or positive.

Abstract

Let $(u_n)_{n \geq 0}$ be a nondegenerate linear recurrence of integers, and let $\mathcal{A}$ be the set of positive integers $n$ such that $u_n$ and $n$ are relatively prime. We prove that $\mathcal{A}$ has an asymptotic density, and that this density is positive unless $(u_n / n)_{n \geq 1}$ is a linear recurrence.

Equations

$$u_n=a_1{u}_{n-1}+a_2{u}_{n-2}+\cdots +a_k{u}_{n-k},$$

$${\mathcal{B}}_u:=\{n\in \mathbf{N}:n\mid u_n\}$$

$$\#{\mathcal{B}}_u(x){\ll }_k\frac{x}{\log x},$$

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

$${\mathcal{B}}_{u,v}:=\{n\in \mathbf{N}:v_n\mid u_n\},$$

$$\#{\mathcal{B}}_{u,v}(x){\ll }_{u,v}x\cdot {\left(\frac{\log \log x}{\log x}\right)}^{h_{u,v}},$$

$$\mathcal{G}:=\{\gcd (n,F_n):n\in \mathbf{N}\}$$

$$\#\mathcal{G}(x)\gg \frac{x}{\log x},$$

$${\mathcal{A}}_u=\{n\in \mathbf{N}:\gcd (n,u_n)=1\}$$

$$f_u(X)=X^k-a_1{X}^{k-1}-a_2{X}^{k-2}-\cdots -a_k.$$

$$u_n=\sum _{i=1}^r{g}_i(n){\alpha }_i^n,$$

$$g_1(0)=\cdots =g_r(0)=0.$$

$$\frac{B_u{u}_n}{n}=\sum _{i=1}^r\frac{B_u{g}_i(n)}{n}{\alpha }_i^n$$

$$D_u(x_1,\dots ,x_k):=\det ({\alpha }_i^{x_j}{)}_{1\le i,j\le k},$$

$$\prod _{1\le x_2,\dots ,x_k\le T}\max \left\{1,|N_{\mathbf{K}}(D_u(0,x_2,\dots ,x_k))|\right\},$$

$${\mathcal{P}}_{u,\gamma }:=\{p:p\nmid a_k,T_u(p)<p^{\gamma }\}.$$

$$\#{\mathcal{P}}_{u,\gamma }(x){\ll }_u\frac{x^{k\gamma }}{\gamma \log x}.$$

$$O_k\left(\frac{x}{T_u(p)}+1\right).$$

$$v_n:=B_u\sum _{i=1}^r\frac{g_i(n)-g_i(0)}{n}{\alpha }_i^n\text{and}{w}_n:=B_u\sum _{i=1}^r{g}_i(0){\alpha }_i^n.$$

$$n{v}_n+w_n=B_u{u}_n=\sigma (B_u{u}_n)=\sigma (n{v}_n+w_n)=n\sigma (v_n)+\sigma (w_n),$$

$${\mathcal{C}}_{u,y}^{-}$$

$${\mathcal{C}}_{u,y}^{+}$$

$$\delta :=\underset{y\to +\infty }{\lim }{\delta }_y$$

$$O\left(\frac{x}{p{T}_w(p)}+1\right).$$

$$\#{\mathcal{C}}_{u,y}^{+}(x)\ll \sum _{y<p\le x}\left(\frac{x}{p{T}_w(p)}+1\right)\ll x\cdot \left(\sum _{p>y}\frac{1}{p{T}_w(p)}+\frac{1}{\log x}\right),$$

$$\sum _{\begin{array}{c}p>y\\ p\in {\mathcal{P}}_{w,\gamma }\end{array}}\frac{1}{p{T}_w(p)}\le \sum _{\begin{array}{c}p>y\\ p\in {\mathcal{P}}_{w,\gamma }\end{array}}\frac{1}{p}={\left[\frac{\#{\mathcal{P}}_{w,\gamma }(t)}{t}\right]}_{t=y}^{+\infty }+{\int }_y^{+\infty }\frac{\#{\mathcal{P}}_{w,\gamma }(t)}{t^2}\mathrm{d}t\ll \frac{1}{y^{1-k\gamma }}=\frac{1}{y^{\gamma }}.$$

$$\sum _{\begin{array}{c}p>y\\ p\notin {\mathcal{P}}_{w,\gamma }\end{array}}\frac{1}{p{T}_w(p)}\le \sum _{\begin{array}{c}p>y\\ p\notin {\mathcal{P}}_{w,\gamma }\end{array}}\frac{1}{p^{1+\gamma }}\ll {\int }_y^{+\infty }\frac{\mathrm{d}t}{t^{1+\gamma }}\ll \frac{1}{y^{\gamma }}$$

$$\frac{\#{\mathcal{C}}_{u,y}^{+}(x)}{x}\ll \frac{1}{y^{\gamma }}+\frac{1}{\log x},$$

$$\underset{x\to +\infty }{\mathrm{limsup}}\left|\frac{\#{\mathcal{C}}_u(x)}{x}-{\delta }_y\right|=\underset{x\to +\infty }{\mathrm{limsup}}\left|\frac{\#{\mathcal{C}}_u(x)}{x}-\frac{\#{\mathcal{C}}_{u,y}^{-}(x)}{x}\right|=\underset{x\to +\infty }{\mathrm{limsup}}\frac{\#{\mathcal{C}}_{u,y}^{+}(x)}{x}\ll \frac{1}{y^{\gamma }},$$

$${\mathcal{C}}_{u,y}^{-}\subseteq \{n\in \mathbf{N}:p\mid n\text{ for some }p\le y\},$$

$$\underset{x\to +\infty }{\mathrm{limsup}}\frac{\#{\mathcal{C}}_{u,y}^{-}(x)}{x}\le 1-\prod _{p\le y}\left(1-\frac{1}{p}\right)\le 1-\frac{c_1}{\log y},$$

Full text

1. Introduction

Let $(u_{n})_{n\geq 0}$ be a linear recurrence over the integers, that is, $(u_{n})_{n\geq 0}$ is a sequence of integers satisfying

[TABLE]

for all integers $n\geq k$, where $a_{1},\ldots,a_{k}\in\mathbf{Z}$ and $a_{k}\neq 0$. To avoid trivialities, we assume that $(u_{n})_{n\geq 0}$ is not identically zero. We refer the reader to [4, Ch. 1-8] for the general terminology and theory of linear recurrences.

The set

[TABLE]

has been studied by several researchers. Assuming that $(u_{n})_{n\geq 0}$ is nondegenerate and that its characteristic polynomial has only simple roots, Alba González, Luca, Pomerance, and Shparlinski [1, Theorem 1.1] proved that

[TABLE]

for all sufficiently large $x>1$. André-Jeannin [2] and Somer [9] studied the arithmetic properties of the elements of $\mathcal{B}_{u}$ when $(u_{n})_{n\geq 0}$ is a Lucas sequence, that is, $(u_{0},u_{1},k)=(0,1,2)$. In such a case, generalizing a previous result of Luca and Tron [6], Sanna [8] proved the upper bound

[TABLE]

as $x\to+\infty$, where the $o(1)$ depends on $a_{1}$ and $a_{2}$. Furthermore, Corvaja and Zannier [3] studied the more general set

[TABLE]

where $(v_{n})_{n\geq 0}$ is another linear recurrence over the integers. Under some mild hypotheses on $(u_{n})_{n\geq 0}$ and $(v_{n})_{n\geq 0}$, they proved that $\mathcal{B}_{u,v}$ has zero asymptotic density [3, Corollary 2], while Sanna [7] gave the bound

[TABLE]

for all $x\geq 3$, where $h_{u,v}$ is a positive integer depending only on $(u_{n})_{n\geq 0}$ and $(v_{n})_{n\geq 0}$.

If $(F_{n})_{n\geq 0}$ is the sequence of Fibonacci numbers, Leonetti and Sanna [5] showed that the set

[TABLE]

has zero asymptotic density, and that

[TABLE]

for all $x\geq 2$.

On the other hand, the set

[TABLE]

does not seem to have attracted so much attention. We prove the following result:

Theorem 1.1**.**

For any nondegenerate linear recurrence $(u_{n})_{n\geq 0}$, the asymptotic density $\mathbf{d}(\mathcal{A}_{u})$ of $\mathcal{A}_{u}$ exists. Moreover, if $(u_{n}/n)_{n\geq 1}$ is not a linear recurrence then $\mathbf{d}(\mathcal{A}_{u})>0$. Otherwise, $\mathcal{A}_{u}$ is finite and, a fortiori, $\mathbf{d}(\mathcal{A}_{u})=0$.

We remark that given the initial conditions and the coefficients of a linear recurrence $(u_{n})_{n\geq 0}$, it is easy to test effectively if $(u_{n}/n)_{n\geq 1}$ is a linear recurrence or not (see Lemma 2.1, in §2).

Notation

Throughout, the letter $p$ always denotes a prime number. For a set of positive integers $\mathcal{S}$, we put $\mathcal{S}(x):=\mathcal{S}\cap[1,x]$ for all $x\geq 1$, and we recall that the asymptotic density $\mathbf{d}(\mathcal{S})$ of $\mathcal{S}$ is defined as the limit of the ratio $\#\mathcal{S}(x)/x$ as $x\to+\infty$, whenever this exists. We employ the Landau–Bachmann “Big Oh” and “little oh” notations $O$ and $o$, as well as the associated Vinogradov symbols $\ll$ and $\gg$, with their usual meanings. Any dependence of the implied constants is explicitly stated or indicated with subscripts.

2. Preliminaries

In this section we give some definitions and collect some preliminary results needed in the later proofs. Let $f_{u}$ be the characteristic polynomial of $(u_{n})_{n\geq 0}$, i.e.,

[TABLE]

Moreover, let $\mathbf{K}$ be the splitting field of $f_{u}$ over $\mathbf{Q}$, and let $\alpha_{1},\ldots,\alpha_{r}\in\mathbf{K}$ be all the distinct roots of $f_{u}$. It is well known that there exist $g_{1},\ldots,g_{r}\in\mathbf{K}[X]$ such that

[TABLE]

for all integers $n\geq 0$. We define $B_{u}$ as the smallest positive integer such that all the coefficients of the polynomials $B_{u}g_{1},\ldots,B_{u}g_{r}$ are algebraic integers.

We have the following easy lemma.

Lemma 2.1**.**

$\{u_{n}/n\}_{n\geq 1}$* is a linear recurrence if and only if*

[TABLE]

In such a case, $\mathcal{A}_{u}$ is finite.

Proof.

The first part of the lemma follows immediately from the fact that any linear recurrence can be written as a generalized power sum like (1) in a unique way (assuming the roots $\alpha_{1},\ldots,\alpha_{r}$ are distinct, and up to the order of the addends). For the second part, if (2) holds then for all positive integer $n$ we have that

[TABLE]

is both a rational number and an algebraic integer, hence it is an integer. Therefore, $n\mid B_{u}u_{n}$, and so $\gcd(n,u_{n})=1$ only if $n\mid B_{u}$, which in turn implies that $\mathcal{A}_{u}$ is finite. ∎

For the rest of this section, we assume that $(u_{n})_{n\geq 0}$ is nondegenerate and that $f_{u}$ has only simple roots, hence, in particular, $r=k$. We write $\Delta_{u}$ for the discriminant of the polynomial $f_{u}$, and we recall that $\Delta_{u}$ is a nonzero integer. If $k\geq 2$, then for all integers $x_{1},\ldots,x_{k}$ we set

[TABLE]

and for any prime number $p$ not dividing $a_{k}$ we define $T_{u}(p)$ as the greatest integer $T\geq 0$ such that $p$ does not divide

[TABLE]

where $N_{\mathbf{K}}(\alpha)$ denotes the norm of $\alpha\in\mathbf{K}$ over $\mathbf{Q}$, and the empty product is equal to $1$. It is known that such $T$ exists [4, p. 88]. If $k=1$, then we set $T_{u}(p):=+\infty$ for all prime numbers $p$ not dividing $a_{1}$. Note that $T_{u}(p)=0$ if and only if $k=2$ and $p$ divides $\Delta_{u}$.

Finally, for all $\gamma\in{]0,1[}$, we define

[TABLE]

We are ready to state two important lemmas regarding $T_{u}(p)$ [1, Lemma 2.1, Lemma 2.2].

Lemma 2.2**.**

For all $\gamma\in{]0,1[}$ and $x\geq 2^{1/\gamma}$ we have

[TABLE]

Lemma 2.3**.**

Assume that $p$ is a prime number not dividing $a_{k}B_{u}\Delta_{u}$ and relatively prime with at least one term of $(u_{n})_{n\geq 0}$. Then, for all $x\geq 1$, the number of positive integers $m\leq x$ such that $u_{pm}\equiv 0\pmod{p}$ is

[TABLE]

Actually, in [1] both Lemma 2.2 and Lemma 2.3 were proved only for $k\geq 2$. However, one can easily check that they are true also for $k=1$.

3. Proof of Theorem 1.1

For all integers $n\geq 0$, define

[TABLE]

Note that both $(v_{n})_{n\geq 0}$ and $(w_{n})_{n\geq 0}$ are linear recurrences of algebraic integers, and that the characteristic polynomial of $(w_{n})_{n\geq 0}$ has only simple roots.

Let $\mathcal{G}$ be the Galois group of $\mathbf{K}$ over $\mathbf{Q}$. Since $u_{n}$ is an integer, for any $\sigma\in\mathcal{G}$ we have that

[TABLE]

for all integers $n\geq 0$. In (3) note that both $n\sigma(v_{n})$ and $\sigma(w_{n})$ are linear recurrences, and the first is a multiple of $n$, while the characteristic polynomial of the second has only simple roots. Since the expression of a linear recurrence as a generalized power sum is unique, from (3) we get that $w_{n}=\sigma(w_{n})$ for any $\sigma\in\mathcal{G}$, hence $w_{n}$ is an integer.

Thanks to Lemma 2.1, we know that $(w_{n})_{n\geq 0}$ is identically zero if and only if $(u_{n}/n)_{n\geq 1}$ is a linear recurrence, and in such a case $\mathcal{A}_{u}$ is finite, so that the claim of Theorem 1.1 is obvious. Hence, we assume that $(w_{n})_{n\geq 0}$ is not identically zero.

For the sake of convenience, put $\mathcal{C}_{u}:=\mathbf{N}\setminus\mathcal{A}_{u}$. Thus we have to prove that the asymptotic density of $\mathcal{C}_{u}$ exists and is less than $1$. For each $y>0$, we split $\mathcal{C}_{u}$ into two subsets:

[TABLE]

It is well known that $(u_{n})_{n\geq 0}$ is definitively periodic modulo $p$, for any prime number $p$. Therefore, it is easy to see that $\mathcal{C}_{u,y}^{-}$ is an union of finitely many arithmetic progressions and a finite subset of $\mathbf{N}$. In particular, $\mathcal{C}_{u,y}^{-}$ has an asymptotic density. If we put $\delta_{y}:=\mathbf{d}(\mathcal{C}_{u,y}^{-})$, then it is clear that $\delta_{y}$ is a bounded nondecreasing function of $y$, hence the limit

[TABLE]

exists finite. We shall prove that $\mathcal{C}_{u}$ has asymptotic density $\delta$. Hereafter, all the implied constants may depend on $(u_{n})_{n\geq 0}$ and $k$. If $n\in\mathcal{C}_{u,y}^{+}(x)$ then there exists a prime $p>y$ such that $p\mid n$ and $p\mid u_{n}$. Furthermore, $B_{u}u_{n}=nv_{n}+w_{n}$ implies that $p\mid w_{n}$. Hence, we can write $n=pm$ for some positive integer $m\leq x/p$ such that $w_{pm}\equiv 0\pmod{p}$. For sufficiently large $y$, we have that $p$ does not divide $f_{w}(0)B_{w}\Delta_{w}$ and is coprime with at least one term of $(w_{s})_{s\geq 0}$, since $(w_{s})_{s\geq 0}$ is not identically zero.

Therefore, by applying Lemma 2.3 to $(w_{s})_{s\geq 0}$, we get that the number of possible values of $m$ is at most

[TABLE]

As a consequence,

[TABLE]

where we also used the Chebyshev’s bound for the number of primes not exceeding $x$. Setting $\gamma:=1/(k+1)$, by partial summation and Lemma 2.2, we have

[TABLE]

On the other hand,

[TABLE]

Thus, putting together (5), (6), and (7), we obtain

[TABLE]

so that

[TABLE]

hence, by letting $y\to+\infty$ in (8) and by using (4), we get that $\mathcal{C}_{u}$ has asymptotic density $\delta$.

It remains only to prove that $\delta<1$. Clearly,

[TABLE]

so that, by Eratosthenes’ sieve and Mertens’ third theorem [10, Ch. I.1, Theorem 11], we have

[TABLE]

for all $y\geq 2$, where $c_{1}>0$ is an absolute constant. Hence, putting together (9) and the last part of (8), we get

[TABLE]

for all $y\geq 2$, where $c_{2}>0$ is an absolute constant.

Finally, picking a sufficiently large $y$, depending on $c_{1}$ and $c_{2}$, the bound (10) yields $\delta<1$. The proof of Theorem 1.1 is complete.