Related papers: On prime values of cyclotomic polynomials

Let $A_n$ denote the height of cyclotomic polynomial $\Phi_n$, where $n$ is a product of $k$ distinct odd primes. We prove that $A_n \le \epsilon_k\phi(n)^{k^{-1}2^{k-1}-1}$ with $-\log\epsilon_k\sim c2^k$, $c>0$. The same statement is true…

In this article, we obtain upper bounds on the number of irreducible factors of some classes of polynomials having integer coefficients, which in particular yield some of the well known irreducibility criteria. For devising our results, we…

Let a(n,k) be the kth coefficient of the nth cyclotomic polynomial. The first two authors showed in part I that if m is a prime power and n and k range over the non-negative integers, then a(mn,k) assumes every integer value. Here this…

Let $\Phi_n(x)$ denote the $n$th cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that $a_n(k)$, the coefficient of $x^k$ in $\Phi_n(x)$, satisfies $|a_n(k)|\le (p+1)/2$ in case $n=pqr$ with $p<q<r$ primes (in this case…

Using the cyclotomic identity we compute sums over d-tuples of monic polynomials in F_q[x] weighted by the multiplicity of their irreducible factors. As consequences we determine explicit expressions for the number of d-tuples of…

The $n^{th}$ cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Its coefficients are the subject of intensive study and some formulas are known for them. Here we are interested in formulas…

We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions…

This paper investigates coefficients of cyclotomic polynomials theoretically and experimentally. We prove the following result. {{\em If $n=p_1\ldots p_k$ where $p_i$ are odd primes and $p_1<p_2<\ldots<p_r<p_1+p_2<p_{r+1}<\ldots<p_t$ with…

Arthur Cohn's irreducibility criterion for polynomials with integer coefficients and its generalization connect primes to irreducibles, and integral bases to the variable $x$. As we follow this link, we find that these polynomials are ready…

It is known that two consecutive coefficients of a ternary cyclotomic polynomial $\Phi_{pqr}(x)=\sum_k a_{pqr}(k)x^k$ differ by at most one. In this paper we give a criterion on $k$ to satisfy $|a_{pqr}(k)-a_{pqr}(k-1)|=1$. We use this to…

We present an elementary identity for the cyclotomic polynomials $\Phi_n(X)$ which reflects a kind of multiplicative property of $\Phi_n(X)$ as a function of $n$, and we explore its connections with the properties of other arithmetical…

Let a_n(k) be the kth coefficient of the nth cyclotomic polynomial Phi_n(x). As n ranges over the integers, a_n(k) assumes only finitely many values. For any such value v we determine the density of integers n such that a_n(k)=v. Also we…

Taking a combinatorial point of view on cyclotomic polynomials leads to a larger class of polynomials we shall call the inclusion-exclusion polynomials. This gives a more appropriate setting for certain types of questions about the…

In this paper we present and analyse a construction of irreducible polynomials over odd prime fields via the transforms which take any polynomial $f \in \mathbf{F}_p[x]$ of positive degree $n$ to $\left(\frac{x}{k} \right)^n \cdot…

Let $q\geqslant 2$ be a fixed prime power. We prove an asymptotic formula for counting the number of monic polynomials that are of degree $n$ and have exactly $k$ irreducible factors over the finite field $\mathbb{F}_q$. We also compare our…

We determine necessary and sufficient conditions for unicritical polynomials to be dynamically irreducible over finite fields. This result extends the results of Boston-Jones and Hamblen-Jones-Madhu regarding the dynamical irreducibility of…

Let $\mathbb{F}_q$ be the finite field with $q$ elements, where $q$ is a prime power and $n$ be a positive integer. In this paper, we explore the factorization of $f(x^{n})$ over $\mathbb{F}_q$, where $f(x)$ is an irreducible polynomial…

We prove a function field analogue of Maynard's result about primes with restricted digits. That is, for certain ranges of parameters n and q, we prove an asymptotic formula for the number of irreducible polynomials of degree n over a…

Mersenne primes and Fermat primes may be thought of as primes of the form $\Phi_m(2)$, where $\Phi_m(x)$ is the $m$th cyclotomic polynomial. This paper discusses the more general problem of primes and composites of this form.

We show that for a random polynomial \[ F(X) = \sum_{n=1}^{N} f(n) X^{n-1}, \] where $f(n)$ is a random completely multiplicative function taking values in $\{\pm 1\}$, one has \[ \limsup_{N \to \infty} \mathbb{P}\big[F(X) \text{ is…