Related papers: Primes in arithmetic progressions and nonprimitive…

Let $\mathbb{F}_p$ be a finite field of size $p$ where $p$ is an odd prime. Let $f(x)\in\mathbb{F}_p[x]$ be a polynomial of positive degree $k$ that is not a $d$-th power in $\mathbb{F}_p[x]$ for all $d\mid p-1$. Furthermore, we require…

Let $K$ be a number field and $G$ a finitely generated torsion-free subgroup of $K^\times$. Given a prime $\mathfrak p$ of $K$ we denote by ${\rm ind}_{\mathfrak p}(G)$ the index of the subgroup $(G\bmod\mathfrak p)$ of the multiplicative…

For a given finitely generated multiplicative subgroup of the rationals which possibly contain negative numbers, we derive, subject to GRH, formulas for the densities of primes for which the index of the reduction group has a given value.…

Let $p=2n+1$ be an odd prime, and let $\zeta_{p^2-1}$ be a primitive $(p^2-1)$-th root of unity in the algebraic closure $\overline{\mathbb{Q}_p}$ of $\mathbb{Q}_p$. We let $g\in\mathbb{Z}_p[\zeta_{p^2-1}]$ be a primitive root modulo…

We define a new congruence relation on the set of integers, leading to a group similar to the multiplicative group of integers modulo $n$. It makes use of a symmetry almost omnipresent in modular multiplications and halves the number of…

This paper presents an analysis of primitive permutation groups of degree $3p$, where $p$ is a prime number, analogous to H. Wielandt's treatment of groups of degree $2p$. It is also intended as an example of the systematic use of…

An ideal setting to exhibit infinite sets of primes $p$ relative to which an integer is a primitive root $\pmod p$ is provided by the B\'ezout subdomain $\widetilde{\mathbb{B}}:=\mathbb{Z}^{\mathbb{P}}/\mathfrak{U}$ of the valuation domain…

In this paper, we introduce the prime trees associated with a finite subset $P$ of the set of all prime numbers, and provide conditions under which the tree is of finite type. Moreover, we compute the density of finite-type subsets $P$. As…

Let k>1 be an integer and let p be a prime. We show that if $p^a\le k<2p^a$ or $k=p^aq+1$ (with 2q<p) for some a=1,2,..., then the set {\binom{n}{k}: n=0,1,2,...} is dense in the ring Z_p of p-adic integers, i.e., it contains a complete…

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…

Let $K$ be a number field and let $G$ be a finitely generated subgroup of $K^\times$. For all but finitely many primes $\mathfrak p$ of $K$, the reduction $(G \bmod \mathfrak p)$ generates a well-defined subgroup of the multiplicative group…

For a polynomial $g(x)$ of deg $k \geq 2$ with integer coefficients and positive integer leading coefficient, we prove an upper bound for the least prime $p$ such that $g(p)$ is in non-homogeneous Beatty sequence $\lbrace \lfloor \alpha…

Rubinstein and Sarnak have shown, conditional on the Riemann hypothesis (RH) and the linear independence hypothesis (LI) on the non-real zeros of $\zeta(s)$, that the set of real numbers $x\ge2$ for which $\pi(x)>$ li$(x)$ has a logarithmic…

For a fixed rational number g and integers a and d the sets N_g(a,d), respectively R_g(a,d), of primes p for which the order, respectively the index of g(mod p) is congruent to a(mod d), are considered. Under the Generalized Riemann…

Let $(b_n) = (b_1, b_2, ...)$ be a sequence of integers. A primitive prime divisor of a term $b_k$ is a prime which divides $b_k$ but does not divide any of the previous terms of the sequence. A zero orbit of a polynomial $f(z)$ is a…

We prove that if p is a prime with a primitive root 2 then S_p(2^p)=p and give a sufficient condition for an equality of kind S_p(2^p)=+or-p.

We prove that the sumset {p^2+b^2+2^n: p is prime and b,n\in N} has positive lower density. We also construct a residue class with odd modulo, which contains no integer of the form p^2+b^2+2^n. And similar results are established for the…

We prove that if a polynomial has a root mod $p$ for every large prime $p$, then it has a real root. As an application, we show that the primes can't be covered by finitely many positive definite binary quadratic forms.

For a prime $p$ and an integer $u$ with $\gcd(u,p)=1$, we define Fermat quotients by the conditions $$ q_p(u) \equiv \frac{u^{p-1} -1}{p} \pmod p, \qquad 0 \le q_p(u) \le p-1. $$ D. R. Heath-Brown has given a bound of exponential sums with…

Let $t \in \mathbb{N}$, $\eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q \leq x^{5/12-\eta}$, $q$ not a multiple of the conductor of the exceptional character $\chi^*$ (if it exists).…