442 research outputs found

### Distribution of integral values for the ratio of two linear recurrences

Let $F$ and $G$ be linear recurrences over a number field $\mathbb{K}$, and
let $\mathfrak{R}$ be a finitely generated subring of $\mathbb{K}$.
Furthermore, let $\mathcal{N}$ be the set of positive integers $n$ such that
$G(n) \neq 0$ and $F(n) / G(n) \in \mathfrak{R}$. Under mild hypothesis,
Corvaja and Zannier proved that $\mathcal{N}$ has zero asymptotic density. We
prove that $\#(\mathcal{N} \cap [1, x]) \ll x \cdot (\log\log x / \log x)^h$
for all $x \geq 3$, where $h$ is a positive integer that can be computed in
terms of $F$ and $G$. Assuming the Hardy-Littlewood $k$-tuple conjecture, our
result is optimal except for the term $\log \log x$

### On the sum of digits of the factorial

Let b > 1 be an integer and denote by s_b(m) the sum of the digits of the
positive integer m when is written in base b. We prove that s_b(n!) > C_b log n
log log log n for each integer n > e, where C_b is a positive constant
depending only on b. This improves of a factor log log log n a previous lower
bound for s_b(n!) given by Luca. We prove also the same inequality but with n!
replaced by the least common multiple of 1,2,...,n.Comment: 4 page

### Covering an arithmetic progression with geometric progressions and vice versa

We show that there exists a positive constant C such that the following
holds: Given an infinite arithmetic progression A of real numbers and a
sufficiently large integer n (depending on A), there needs at least Cn
geometric progressions to cover the first n terms of A. A similar result is
presented, with the role of arithmetic and geometric progressions reversed.Comment: 4 page

### A note on primes in certain residue classes

Given positive integers $a_1,\ldots,a_k$, we prove that the set of primes $p$
such that $p \not\equiv 1 \bmod{a_i}$ for $i=1,\ldots,k$ admits asymptotic
density relative to the set of all primes which is at least $\prod_{i=1}^k
\left(1-\frac{1}{\varphi(a_i)}\right)$, where $\varphi$ is the Euler's totient
function. This result is similar to the one of Heilbronn and Rohrbach, which
says that the set of positive integer $n$ such that $n \not\equiv 0 \bmod a_i$
for $i=1,\ldots,k$ admits asymptotic density which is at least $\prod_{i=1}^k
\left(1-\frac{1}{a_i}\right)$

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

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 $F_k$

### A coprimality condition on consecutive values of polynomials

Let $f\in\mathbb{Z}[X]$ be quadratic or cubic polynomial. We prove that there
exists an integer $G_f\geq 2$ such that for every integer $k\geq G_f$ one can
find infinitely many integers $n\geq 0$ with the property that none of
$f(n+1),f(n+2),\dots,f(n+k)$ is coprime to all the others. This extends
previous results on linear polynomials and, in particular, on consecutive
integers

### The density of numbers $n$ having a prescribed G.C.D. with the $n$th Fibonacci number

For each positive integer $k$, let $\mathscr{A}_k$ be the set of all positive
integers $n$ such that $\gcd(n, F_n) = k$, where $F_n$ denotes the $n$th
Fibonacci number. We prove that the asymptotic density of $\mathscr{A}_k$
exists and is equal to $\sum_{d = 1}^\infty
\frac{\mu(d)}{\operatorname{lcm}(dk, z(dk))}$ where $\mu$ is the M\"obius
function and $z(m)$ denotes the least positive integer $n$ such that $m$
divides $F_n$. We also give an effective criterion to establish when the
asymptotic density of $\mathscr{A}_k$ is zero and we show that this is the case
if and only if $\mathscr{A}_k$ is empty

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