Related papers: Primitive Divisors of some Lehmer-Pierce Sequences

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements. For a positive divisor $r$ of $q^n-1$, the element $\alpha \in \mathbb{F}_{q^n}^*$ is called \textit{$r$-primitive} if its multiplicative order is $(q^n-1)/r$. Also, for a…

Let H_q(S_n) be the Iwahori-Hecke algebra of the symmetric group. This algebra is semisimple over the rational function field Q(q), where q is an indeterminate, and its irreducible representations over this field are q-analogues S_q(lambda)…

A number field $K$ is called primitive if $\mathbb Q$ and $K$ are the only subfields of $K$. Let $X$ be a nice curve over $\mathbb Q$ of genus $g$. A point $P$ of degree $d$ on $X$ is called primitive if the field of definition $\mathbb…

In this paper, we obtain analogues of Zsigmondy's theorem and the primitive divisor results for the Lucas and Lehmer sequences in polynomial rings of several variables.

Let $\mathbb{K}$ be a number field of degree $n$ over $\mathbb{Q}$. Let $\widehat{\mathbb{A}}$ be the set of integers of $\mathbb{K}$ which are primitive over $\mathbb{Q}$ and $I(\mathbb{K})$ be its index. Gunji and McQuillan defined the…

A positive integer n is called a covering number if there are some distinct divisors n_1,...,n_k of n greater than one and some integers a_1,...,a_k such that Z is the union of the residue classes a_1(mod n_1),...,a_k(mod n_k). A covering…

Let K be a number field, let f(x) in K(x) be a rational function of degree d> 1, and let z in K be a wandering point such that f^n(z) is nonzero for all n > 0. We prove that if the abc-conjecture holds for K, then for all but finitely many…

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erd\H{o}s proved in 1935 that the weighted sum $\sum1/(n \log n)$ for $n$ ranging over a primitive set $A$ is universally bounded over all choices for…

We prove the existence of primitive sets (sets of integers in which no element divides another) in which the gap between any two consecutive terms is substantially smaller than the best known upper bound for the gaps in the sequence of…

Let $r$ be a positive divisor of $q-1$ and $f(x,y)$ a rational function of degree sum $d$ over $\mathbb{F}_q$ with some restrictions, where the degree sum of a rational function $f(x,y) = f_1(x,y)/f_2(x,y)$ is the sum of the degrees of…

Let $R$ be a commutative ring and $n\geq1$ and $p\geq0$ two integers. Let $h_{k,\ i}$ be an element of $R$ for all $k\in\mathbb Z$ and $i\in [n]$. For any $\alpha\in\mathbb Z^n$, we define \[ t_{\alpha}:=\det\begin{pmatrix} h_{\alpha_1+1,\…

The previous paper [4] proved the existence of primitive polynomials and primitive normal polynomials of degree n with k prescribed coefficients in the finite field GF(q) for all sufficiently large q. This paper presents a loger versions of…

In this paper, we show that the exponent set of nonnegative primitive tensors with order m(\geq 3) and dimension n is {1,2,\ldots, (n-1)^2+1}; and propose some open problems for further research.

The divisor sequence of an irreducible element (\textit{atom}) $a$ of a reduced monoid $H$ is the sequence $(s_n)_{n\in \mathbb{N}}$ where, for each positive integer $n$, $s_n$ denotes the number of distinct irreducible divisors of $a^n$.…

The discriminant of a polynomial of the form $\pm x^n \pm x^m \pm 1$ has the form $n^n \pm m^m(n-m)^{n-m}$ when $n,m$ are relatively prime. We investigate when these discriminants have prime power divisors. We explain several symmetries…

The appearance of primes in a family of linear recurrence sequences labelled by a positive integer $n$ is considered. The terms of each sequence correspond to a particular class of Lehmer numbers, or (viewing them as polynomials in $n$)…

Let $\theta$ be an algebraic integer and $f(x)=x^{n}+ax^{n-1}+bx+c$ be the minimal polynomial of $\theta$ over the rationals. Let $K=\mathbb{Q}(\theta)$ be a number field and $\mathcal{O}_{K}$ be the ring of integers of $K.$ In this…

The discriminator of an integer sequence $\textbf{s} = (s(i))_{i \geq 0}$, introduced by Arnold, Benkoski, and McCabe in 1985, is the function $D_{\textbf{s}} (n)$ that sends $n$ to the least integer $m$ such that the numbers $s(0), s(1),…

Let $k$ be a field and let $R$ be a countable dimensional prime von Neumann regular $k$-algebra. We show that $R$ is primitive, answering a special case of a question of Kaplansky.

A set of integers greater than 1 is primitive if no element divides another. Erd\H{o}s proved in 1935 that the sum of $1/(n \log n)$ for $n$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked…