Related papers: Optimal primitive sets with restricted primes

For $p$ being a large prime number, and $A \subset \mathbb{F}_p$ we prove the following: $(i)$ If $A(A+A)$ does not cover all nonzero residues in $\mathbb{F}_p$, then $|A| < p/8 + o(p)$. $(ii)$ If $A$ is both sum-free and satisfies $A =…

Let $p$ be a prime number. We say that a positive integer $n$ is a Sylow $p$-number if there exists a finite group having exactly $n$ Sylow $p$-subgroups. When $p=2$, every odd integer is a Sylow $2$-number. In contrast, when $p$ is odd,…

Let $p>1$ be a large prime number and let $x=O((\log p)^2(\log\log p)^5$ be a real number. It is proved that the least consecutive pair of primitive roots $u\ne\pm1, v^2$ and $u+1$ satisfies the upper bound $u\ll x$ in the prime field…

We say a natural number~$n$ is abundant if $\sigma(n)>2n$, where $\sigma(n)$ denotes the sum of the divisors of~$n$. The aliquot parts of~$n$ are those divisors less than~$n$, and we say that an abundant number~$n$ is pseudoperfect if there…

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.

The greatest power of a prime $p$ dividing the natural number $n$ will be denoted by $n_p$. Let $Ind_G(g)=|G:C_G(g)|$. Suppose that $G$ is a finite group and $p$ is a prime. We prove that if there exists an integer $\alpha>0$ such that…

Using a sieve-theoretic argument, we show that almost all gaps $(p_n, p_{n+1})$ between consecutive primes $p_n, p_{n+1}$ contain a natural number $m$ whose least prime factor $p(m)$ is at least the length $p_{n+1} - p_n$ of the gap,…

Let $X$ be a scheme of finite type over $\mathbf{Z}$. For $p \in \mathcal{P}$ the set of prime numbers, let $N_{X}(p)$ be the number of $\mathbf{F}_{p}$-points of $X/\mathbf{F}_{p}$. For fixed $n\geq 1$ and $a_{1}, \ldots, a_{n} \in…

Let $p$ be a prime number, and let $S$ be the numerical semigroup generated by the prime numbers not less than $p$. We compare the orders of magnitude of some invariants of $S$ with each other, e. g., the biggest atom $u$ of $S$ with $p$…

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.…

Let $P$ and $T$ be disjoint sets of prime numbers with $T$ finite. A simple formula is given for the natural density of the set of square-free numbers which are divisible by all of the primes in $T$ and by none of the primes in $P$. If $P$…

ABSTRACT. In this article we present a point of view that highlights the importance of finding the upper bounds for prime gaps, in order to solve the twin primes conjecture and the Goldbach conjecture. For this purpose, we present a…

A famous conjecture of Artin asserts that any integer $a$ that is neither $-1$ nor a square should be a primitive root (mod $p$) for a positive proportion of primes $p$. Moreover, using a heuristic argument, Artin guessed an explicit…

This monograph considers a few topics in the theory of primitive roots g(p) modulo a prime p>=2. A few estimates of the least primitive roots g(p) and the least prime primitive roots g^*(p) modulo p, a large prime, are determined. One of…

Let a and b be non-zero rational numbers that are multiplicatively independent. We study the natural density of the set of primes p for which the subgroup of the multiplicative group of the finite field with p elements generated by (a\mod…

A primorial prime is a prime number of the form $p_n\# \pm 1$ where $p_n\#$ denotes the product of all primes less than or equal to $p_{n}$, the $n$-th prime. We show that the probability along the lines of Mertens' Theorem that either…

Let $H(n) = \prod_{p|n}\frac{p}{p-1}$ where $p$ ranges over the primes which divide $n$. It is well known that if $n$ is a primitive non-deficient number, then $H(n) > 2$. We examine inequalities of the form $H(n)> 2 + f(n)$ for various…

We consider a finite group $G$ with a normal subgroup $N$ so that all elements of $G \setminus N$ have prime power order. We prove that if there is a prime $p$ so that all the elements in $G \setminus N$ have $p$-power order, then either…

We consider the sums of the form $$ S=\sum_{x=1}^{N} \exp\big((ax+b_1g_1^x+... +b_rg_r^x)/p \big) $$, where $p$ is prime and $g_1,..., g_r$ are primitive roots $\pmod p$. An almost forty years old problem of L. J. Mordell asks to find a…

Let $A$ be a subset of the cyclic group $\mathbf{Z}/p\mathbf{Z}$ with $p$ prime. It is a well-studied problem to determine how small $|A|$ can be if there is no unique sum in $A+A$, meaning that for every two elements $a_1,a_2\in A$, there…