On the distribution of numbers related to the divisors of $x^n-1$

In a recent paper, we considered integers n for which the polynomial x^n - 1 has a divisor in Z[x] of every degree up to n, and we gave upper and lower bounds for their distribution. In this paper, we consider those n for which the polynomial x^n-1 has a divisor in F_p[x] of every degree up to n, where p is a rational prime. Assuming the validity of the Generalized Riemann Hypothesis, we show that such integers n have asymptotic density 0.

In this paper, we examine a natural question concerning the divisors of the polynomial x^n-1: "How often does x^n-1 have a divisor of every degree between 1 and n?" In a previous paper, we considered the situation when x^n-1 is factored in Z[x]. In this paper, we replace Z[x] with F_p[x], where p is an arbitrary-but-fixed prime. We also consider those n where this condition holds for all p.

Let $a(r,n)$ be $r$th coefficient of $n$th cyclotomic polynomial. Suzuki proved that $\{a(r,n)|r\geq 1,n\geq 1\}=\mathbb{Z}$. If $m$ and $n$ are two natural numbers we prove an analogue of Suzuki's theorem for divisors of $x^n-1$ with exactly $m$ irreducible factors. We prove that for every finite sequence of integers $n_1,\ldots,n_r$ there exists a divisor $f(x)=\sum_{i=0}^{deg(f)}c_ix^i$ of $x^n-1$ for some $n\in \mathbb{N}$ such that $c_i=n_i$ for $1\leq i \leq r$. Let $H(...

A positive integer $n$ is called $\varphi$-practical if the polynomial $X^n-1$ has a divisor in $\mathbb{Z}[X]$ of every degree up to $n$. In this paper, we show that the count of $\varphi$-practical numbers in $[1, x]$ is asymptotic to $C x/\log x$ for some positive constant $C$ as $x \rightarrow \infty$.

The author \cite{4} proved that, for every set $S$ of positive integers containing 1 (finite or infinite) there exists the density $h=h(E(S))$ of the set $E(S)$ of numbers whose prime factorizations contain exponents only from $S,$ and gave an explicit formula for $h(E(S)).$ In this paper we give an equivalent polynomial formula for $\log h(E(S))$ which allows to get a fast calculation of $h(E(S)).$

Let $r,n$ be two natural numbers and let $H(r,n)$ denote the maximal absolute value of $r$th coefficient of divisors of $x^n-1$. In this paper, we show that $\sum_{n\leq x}H(r,n)$ is asymptotically equal to $c(r)x(\log x)^{2^r-1}$ for some constant $c(r)>0$. Furthermore, we give an explicit expression of $c(r)$ in terms of $r$.

Fix a field $F$. In this paper, we study the sets $\D_F(n) \subset [0,n]$ defined by [\D_F(n):= {0 \leq m \leq n: T^n-1\text{has a divisor of degree $m$ in} F[T]}.] When $\D_F(n)$ consists of all integers $m$ with $0 \leq m \leq n$, so that $T^n-1$ has a divisor of every degree, we call $n$ an $F$-practical number. The terminology here is suggested by an analogy with the practical numbers of Srinivasan, which are numbers $n$ for which every integer $0 \leq m \leq \sigma(n)$ c...

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]...

Let $\phi(n)$denote Euler's phi function. We study the distribution of the numbers $gcd(n,\phi(n))$ and their divisors. Our results generalize previous results of Erd\"{o}s and Pollack.

In this paper we prove that polynomials $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$ of degree $d \geq 3$, satisfying certain hypotheses, take on the expected density of $(d-1)$-free values. This extends the authors' earlier result where a different method implied the similar statement for polynomials of degree $d\geq 5$.