Related papers: On the density of primes in arithmetic progression…

We estimate the asymptotic density of the set $\bar{A}$ of primes $p$ satisfying the constraint that $p+1$ and $p-1$ have only one prime divisor larger than $3$. We also estimate the density of a maximal subset $\bar{B} \subset \bar{A}$…

Let $F$ be an integral linear recurrence, $G$ be an integer-valued polynomial splitting over the rationals, and $h$ be a positive integer. Also, let $\mathcal{A}_{F,G,h}$ be the set of all natural numbers $n$ such that $\gcd(F(n), G(n)) =…

This paper is devoted to the theory of prime numbers. In this paper we first introduce the notion of a matrix of prime numbers. Then, in order to investigate the density of prime numbers in separate rows of the matrix under consideration,…

This note presents an upper bound for the least prime primitive roots $g^*(p)$ modulo $p$, a large prime. The current literature has several estimates of the least prime primitive root $g^*(p)$ modulo a prime $p\geq 2$ such as $g^*(p)\ll…

We consider the representation of primes as a sum of a prime and twice a triangular number. We prove that a subset of the primes having density 1 is expressible in this form. We conjecture that every odd prime number is expressible as a sum…

Let P denote the set of all primes. Suppose that P_1, P_2, P_3 are three subsets of P with the sum of their lower densities relative to P is greater than 2. We prove that for sufficiently large odd integer n, there exist p_i\in P_i such…

We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi's theorem, which asserts that any subset of the integers of positive density contains progressions of…

We consider integer recurrences of the form a_n = f(a_{n-1}), where f is a quadratic polynomial with integer coefficients. We show, for four infinite families of f, that the set of primes dividing at least one term of such a sequence must…

E. Artin conjectured that any integer $a >1$ which is not a perfect square is a primitive root modulo $p$ for infinitely many primes $p.$ Let $f_a(p)$ be the multiplicative order of the non-square integer $a$ modulo the prime $p.$ M. R.…

In this paper, we show that various kinds of integer polynomials with prescribed properties of their roots have positive density. For example, we prove that almost all integer polynomials have exactly one or two roots with maximal modulus.…

Let $p$ be a prime and let $L$ be a quadratic $\mathbb{Z}_p$-lattice with quadratic form $Q$. For $t\neq 0$ the local representation density $\alpha_p(t;L)$ is the stable normalised growth of the congruence counts of solutions to…

Let $\mathcal{A}$ denote a finite set of arithmetic progressions of positive integers and let $s \geq 2$ be an integer. If the cardinality of $\mathcal{A}$ is at least 2 and $U$ is the union formed by taking certain arithmetic progressions…

Let $k\in\mathbb{N}$ and let $f_1,\ldots,f_k$ belong to a Hardy field. We prove that under some natural conditions on the $k$-tuple $(f_1,\ldots,f_k)$ the density of the set $$ \big\{n\in \mathbb{N}: \text{gcd}(n,\lfloor…

Let $p>1$ be a large prime number, let $q=O(\log\log p)$ and let $1\leq a<q$ be a pair of relatively prime integers. It is proved that there is a prime primitive root $u\ll (\log p)(\log \log p)^5$ such that $u\equiv a\bmod q$ in the prime…

The set of integers which can be written as the sum of four prime cubes has lower density at least $0.009664$. This improves earlier bounds of $0.003125$ by Ren and $0.005776$ by Liu.

Assuming the Generalized Riemann Hypothesis, we prove the following: If b is an integer greater than one, then the multiplicative order of b modulo N is larger than N^(1-\epsilon) for all N in a density one subset of the integers. If A is a…

Let f be a non-CM newform of weight k > 1. Let L be a subfield of the coefficient field of f. We completely settle the question of the density of the set of primes p such that the p-th coefficient of f generates the field L. This density is…

For a given monic integral polynomial $f(x)$ of degree $n$, we define local roots $r_i$ of $f(x)$ for a completely decomposable prime $p$ by $r_i \in \mathbb{Z}$, $f(r_i) \equiv 0 \bmod p$ and $0 \le r_1 \le r_2 \le \dots \le r_n < p$. With…

We describe a straightforward method to generate a random prime q such that the multiplicative group GF(q)* also has a random large prime-order subgroup. The described algorithm also yields this order p as well as a p'th primitive root of…

Let $m$ be any positive integer and let $\delta_1,\delta_2\in\{1,-1\}$. We show that for some constanst $C_m>0$ there are infinitely many integers $n>1$ with $p_{n+m}-p_n\le C_m$ such that $$\left(\frac{p_{n+i}}{p_{n+j}}\right)=\delta_1\…