Related papers: Primes with small primitive roots

Let F be the function field of a curve over a p-adic field. Let D/F be a central division algebra of prime exponent $\ell$ which is different from p. Assume that F contains a primitive ${\ell}^2$-th root of unity. Then the group…

Fix a prime $p \ge 5$ and define $g(2n,p)=\#\{(h,k)\in\mathbb{Z}_{>0}^2 : h+k=2n,\; h\le k,\; \gcd(h,6p)=\gcd(k,6p)=1\}$. We derive explicit closed-form expressions for $g(2n,p)$ in terms of the canonical remainder operator…

Let $G$ be a finite (not necessarily abelian) group and let $p=p(G)$ be the smallest prime number dividing $|G|$. We prove that $d(G)\leq \frac{|G|}{p}+9p^2-10p$, where $d(G)$ denotes the small Davenport constant of $G$ which is defined as…

Let $\chi$ be a Dirichlet character modulo a prime~$p$. We give explicit upper bounds on $q_1<q_2<\dots<q_n$, the $n$ smallest prime nonresidues of $\chi$. More precisely, given $n_0$ and $p_0$ there exists an absolute constant…

We describe a deterministic algorithm for finding a generating element of the multiplicative group of the finite field $\mathbb{F}_{p^n}$ where $p$ is a prime. In time polynomial in $p$ and $n$, the algorithm either outputs an element that…

The probability of finding a prime multiplet, i.e., a sequence of primes $p$ and $p+a_i$, $i=1... m$, being all primes where $p$ is some prime less than the integer $n$ is naively $1/log(n)^{m+1}$. It is shown that, in reality, it is…

For $k\ge1$, a $k$-almost prime is a positive integer with exactly $k$ prime factors, counted with multiplicity. In this article we give elementary proofs of precise asymptotics for the reciprocal sum of $k$-almost primes. Our results match…

Write $T(n)$ as the sum of the reciprocals of the primes which divide $n$. Write $H(n) = \prod_{p|n}p/(p-1)$ where the product is over the prime divisors of $n$. We prove new bounds for $T(n)$ and $H(n)$ in terms of the smallest prime…

The p-rough numbers are those numbers all of whose prime factors are greater than p. These are exactly those numbers left after Eratosthenes sieve has been advanced from 2 through the prime p. Here we show that for fixed p there is a line…

Let $p$ be a prime and $p_1,\ldots, p_r$ be distinct prime divisors of $p-1$. We prove that the smallest positive integer $n$ which is a simultaneous $p_1,\ldots,p_r$-power nonresidue modulo $p$ satisfies $$ n<p^{1/4 -…

Let $[\, \cdot\,]$ be the floor function and $\|x\|$ denotes the distance from $x$ to the nearest integer. In this paper we show that whenever $\alpha$ is irrational and $\beta$ is real then for any fixed $1<c<12/11$ there exist infinitely…

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 $p$ be a prime and let $G$ be a finite group such that the smallest prime that divides $|G|$ is $p$. We find sharp bounds, depending on $p$, for the commuting probability and the average character degree to guarantee that $G$ is…

In this paper we measure how efficiently a finite simple group $G$ is generated by its elements of order $p$, where $p$ is a fixed prime. This measure, known as the $p$-width of $G$, is the minimal $k\in \mathbb{N}$ such that any $g\in G$…

In this work we show that the prime distribution is deterministic. Indeed the set of prime numbers P can be expressed in terms of two subsets of N using three specific selection rules, acting on two sets of prime candidates. The prime…

Let $\{Z_t, t\geq 0\}$ be a strictly stable process on $\R$ with index $\alpha\in (0,2]$. We prove that for every $p > \alpha$, there exists $\gamma = \gamma (\alpha, p)$ and $\k = \k (\alpha, p)\in (0, +\infty)$ such that…

Let b be an odd integer such that b=+/-1 (mod 8) and let q be a prime with primitive root 2 such that q does not divide b. We show that if (p(k)) is a sequence of odd primes, with 0<=k<=q-2 such that p(k)=2p(k-1)+b for all 1<=k<=q-2, then…

For a field $\mathbb{F}$ and integers $d$ and $k$, a set of vectors of $\mathbb{F}^d$ is called $k$-nearly orthogonal if its members are non-self-orthogonal and every $k+1$ of them include an orthogonal pair. We prove that for every prime…

We show that for any fixed $\eps>0$, there are numbers $\delta>0$ and $p_0\ge 2$ with the following property: for every prime $p\ge p_0$ and every integer $N$ such that $p^{1/(4\sqrt{e})+\eps}\le N\le p$, the sequence $1,2,...,N$ contains…

Given an odd prime $p$, we provide formulas for the Hensel lifts of polynomial roots modulo $p$, and give an explicit factorization over the ring of formal power series with integer coefficients for certain reducible polynomials whose…