Related papers: The Least Prime Primitive Root and the Shifted Sie…

We investigate, using the weighted linear sieve, the distribution of almost-primes among the residue classes (mod p) that generate the multiplicative group of reduced residue classes. We are concerned with finding an upper bound for the…

Let $p$ be a prime and let $g(p)$ be the least primitive root modulo $p$. We prove that for any $\epsilon>0$ and $p$ large enough the bound \begin{align} g(p)\ll p^{\frac{1}{4\sqrt{e}}+\epsilon} \nonumber \end{align} holds for most prime…

$ $The aim of this thesis is to lower the bound on square-free primitive roots modulo primes. Let $g^{\Box}(p)$ be the least square-free primitive root modulo $p$. We have proven the following two theorems. Theorem 0.1. $$g^{\Box}(p) <…

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…

Let $q\ne \pm1,v^2$ be a fixed integer, and let $x\geq 1$ be a large number. The least prime number $p \geq3 $ such that $q$ is a primitive root modulo $p$ is conjectured to be $p\ll (\log q)(\log \log q)^3),$ where $\gcd(p,q)=1$. This note…

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…

We use character sum estimates to give a bound on the least square-full primitive root modulo a prime. Specifically, we show that there is a square-full primitive root mod $p$ less than $p^{2/3 + 3/(4 \sqrt{e})+ \epsilon}$, and we give some…

Let $g(p)$ denote the least primitive root modulo $p$, and $h(p)$ the least primitive root modulo $p^2$. We computed $g(p)$ and $h(p)$ for all primes $p\le 10^{16}$. Here we present the results of that computation and prove three theorems…

We consider an analogue of Artin's primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p+1. An extension of…

We give a method for producing explicit bounds on $g(p)$, the least primitive root modulo $p$. Using our method we show that $g(p)<2r\,2^{r\omega(p-1)}\,p^{\frac{1}{4}+\frac{1}{4r}}$ for $p>10^{56}$ where $r\geq 2$ is an integer parameter.…

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…

Assuming the generalized Lindel\"of hypothesis for Dirichlet $L$-functions, we establish that the least prime $p\equiv a\pmod{q}$ satisfies $p\ll_{\varepsilon} q^{2+\varepsilon}$. This achieves a bound that nearly matches the classical…

Let $E/\mathbb{Q}$ be an elliptic curve. For a prime $p$ of good reduction, let $r(E,p)$ be the smallest non-negative integer that gives the $x$-coordinate of a point of maximal order in the group $E(\mathbb{F}_p)$. We prove unconditionally…

We provide an explicit estimate on the least primitive root mod $p^{2}$. We show, in particular, that every prime $p$ has a primitive root mod $p^{2}$ that is less than $p^{0.99}$.

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…

Let $p\geq3$ be a large prime and let $n(p)\geq2$ denotes the least quadratic nonresidue modulo $p$. This note sharpens the standard upper bound of the least quadratic nonresidue from the unconditional upper bound $n(p)\ll…

In this paper we examine Grosswald's conjecture on $g(p)$, the least primitive root modulo $p$. Assuming the Generalized Riemann Hypothesis (GRH), and building on previous work by Cohen, Oliveira e Silva and Trudgian, we resolve Grosswald's…

For a prime $p$, let $c(p)$ denote the limiting fraction of $n\times n$ matrices over $\mathbb{F}_p$ whose determinant is a primitive root modulo $p$. The quantity $c(p)$ is a natural multiplicative deformation of the totient ratio…

In this paper, we prove several results on the structure of maximal sets $S \subseteq [N]$ such that $S$ mod $p$ is contained in a short arithmetic progression, or the union of short progressions, where $p$ ranges over a subset of primes in…

Primitive roots of 1 mod p^k (k>2 and odd prime p) are sought, in cyclic units group G_k = A_k B_k mod p^k, coprime to p, of order (p-1)p^{k-1}. 'Core' subgroup A_k has order p-1 independent of k, and p+1 generates 'extension' subgroup B_k…