24 research outputs found

### Asymptotic behavior of the least common multiple of consecutive reducible quadratic progression terms

Let $l$ and $m$ be two integers with $l>m\ge 0$, and let $f(x)$ be the
product of two linear polynomials with integer coefficients. In this paper, we
show that $\log {\rm lcm}_{mn<i\le ln}\{f(i)\}=An+o(n)$, where $A$ is a
constant depending only on $l$, $m$ and $f$.Comment: 13 page

### The distribution of divisors of polynomials

Let $F(x)$ be an irreducible polynomial with integer coefficients and degree
at least 2. For $x\ge z\ge y\ge 2$, denote by $H_F(x, y, z)$ the number of
integers $n\le x$ such that $F(n)$ has at least one divisor $d$ with $y<d\le
z$. We determine the order of magnitude of $H_F(x, y, z)$ uniformly for
$y+y/\log^C y < z\le y^2$ and $y\le x^{1-\delta}$, showing that the order is
the same as the order of $H(x,y,z)$, the number of positive integers $n\le x$
with a divisor in $(y,z]$. Here $C$ is an arbitrarily large constant and
$\delta>0$ is arbitrarily small.Comment: v2. minor edits and correction

### Uniform lower bound for the least common multiple of a polynomial sequence

Let $n$ be a positive integer and $f(x)$ be a polynomial with nonnegative
integer coefficients. We prove that ${\rm lcm}_{\lceil n/2\rceil \le i\le n}
\{f(i)\}\ge 2^n$ except that $f(x)=x$ and $n=1, 2, 3, 4, 6$ and that $f(x)=x^s$
with $s\ge 2$ being an integer and $n=1$, where $\lceil n/2\rceil$ denotes the
smallest integer which is not less than $n/2$. This improves and extends the
lower bounds obtained by Nair in 1982, Farhi in 2007 and Oon in 2013.Comment: 6 pages. To appear in Comptes Rendus Mathematiqu

### The elementary symmetric functions of a reciprocal polynomial sequence

Erd\"{o}s and Niven proved in 1946 that for any positive integers $m$ and
$d$, there are at most finitely many integers $n$ for which at least one of the
elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ are
integers. Recently, Wang and Hong refined this result by showing that if $n\geq
4$, then none of the elementary symmetric functions of $1/m, 1/(m+d), ...,
1/(m+(n-1)d)$ is an integer for any positive integers $m$ and $d$. Let $f$ be a
polynomial of degree at least $2$ and of nonnegative integer coefficients. In
this paper, we show that none of the elementary symmetric functions of $1/f(1),
1/f(2), ..., 1/f(n)$ is an integer except for $f(x)=x^{m}$ with $m\geq2$ being
an integer and $n=1$.Comment: 4 pages. To appear in Comptes Rendus Mathematiqu

### The least common multiple of consecutive arithmetic progression terms

Let $k\ge 0,a\ge 1$ and $b\ge 0$ be integers. We define the arithmetic
function $g_{k,a,b}$ for any positive integer $n$ by
$g_{k,a,b}(n):=\frac{(b+na)(b+(n+1)a)...(b+(n+k)a)} {{\rm
lcm}(b+na,b+(n+1)a,...,b+(n+k)a)}.$ Letting $a=1$ and $b=0$, then $g_{k,a,b}$
becomes the arithmetic function introduced previously by Farhi. Farhi proved
that $g_{k,1,0}$ is periodic and that $k!$ is a period. Hong and Yang improved
Farhi's period $k!$ to ${\rm lcm}(1,2,...,k)$ and conjectured that $\frac{{\rm
lcm}(1,2,...,k,k+1)}{k+1}$ divides the smallest period of $g_{k,1,0}$.
Recently, Farhi and Kane proved this conjecture and determined the smallest
period of $g_{k,1,0}$. For the general integers $a\ge 1$ and $b\ge 0$, it is
natural to ask the interesting question: Is $g_{k,a,b}$ periodic? If so, then
what is the smallest period of $g_{k,a,b}$? We first show that the arithmetic
function $g_{k,a,b}$ is periodic. Subsequently, we provide detailed $p$-adic
analysis of the periodic function $g_{k,a,b}$. Finally, we determine the
smallest period of $g_{k,a,b}$. Our result extends the Farhi-Kane theorem from
the set of positive integers to general arithmetic progressions.Comment: 10 pages. To appear in Proceedings of the Edinburgh Mathematical
Societ

### Asymptotic behavior of the least common multiple of consecutive arithmetic progression terms

Let $l$ and $m$ be two integers with $l>m\ge 0$, and let $a$ and $b$ be
integers with $a\ge 1$ and $a+b\ge 1$. In this paper, we prove that $\log {\rm
lcm}_{mn<i\le ln}\{ai+b\} =An+o(n)$, where $A$ is a constant depending on $l,
m$ and $a$.Comment: 8 pages. To appear in Archiv der Mathemati

### The least common multiple of a sequence of products of linear polynomials

Let $f(x)$ be the product of several linear polynomials with integer
coefficients. In this paper, we obtain the estimate: $\log {\rm lcm}(f(1), ...,
f(n))\sim An$ as $n\rightarrow\infty$, where $A$ is a constant depending on
$f$.Comment: To appear in Acta Mathematica Hungaric