44 problems

A primitive root modulo a prime $p$ is an integer whose multiplicative order modulo $p$ is $p-1$. Artin's conjecture. The integer $2$ is a primitive root modulo infinitely many pri…

Let $a$ be a rational integer with $a\neq 1$ that is not a perfect square. Consider the set of rational primes $p$ for which $a$ is a primitive root modulo $p$. Artin's primitive r…

Let $p$ be an odd prime, and let $g(p)$ and $h(p)$ denote the least primitive roots modulo $p$ and modulo $p^2$, respectively. Grosswald's conjecture. Grosswald conjectured that ……

Let ${\cal P}_2$ denote the set of primes having $2$ as a primitive root, and let $A$ be the constant defining the density of ${\cal P}_2$. Rodier's conjecture. The density of the…

Let $p>3$ be prime. A primitive root modulo $p$ is an element generating $(\mathbb Z/p\mathbb Z)^*$. Brizolis's conjecture. There exist an integer $x$ and a primitive root…

Let $K$ be a global field and $F$ a finite normal extension of $K$. Let $C$ be a conjugacy-stable subset of $\operatorname{Gal}(F/K)$, let $d$ be a positive integer (coprime to the…

Let $R$ be a second-order linear recurrence whose characteristic polynomial is reducible over $\mathbb Q$, so that, in the notation of the source, its general term has the form…

Let $f$ be a polynomial representing infinitely many distinct primes, let $g\in G$, and let $c_g(f)$ denote the number of primes represented by $f$ for which $g$ is a primitive roo…

Primitive-root density conjecture. The quotient $R_g(f;x)$ tends to a limit as $x$ tends to infinity; denote this conjectural density by $\delta_g(f)$. This predicts that the relat…

Primitive-root conjecture. There exists such an $a$ and infinitely many primes $p$ of $\mathbf Q$ for which the subgroup generated by…

For a prime $p$, let $R_p$ denote the relevant Chebyshev residue structure, and let $\omega_a$ be the element associated with $a\in R_p$. A Chebyshev primitive root is an element…

Let $p$ be a prime, and let $g_s(p)$ denote the least stationary primitive root in $\mathbb{Z}/p\mathbb{Z}$. Let $\varepsilon>0$ be a small number. Least stationary primitive root…

Golomb's generalized primitive root conjecture. For every such $a$ and $r$, there are infinitely many primes $p$ satisfying

Let $m$ be an integer that is neither a square nor $-1$, and let $a,b\in\mathbb{Z}$ satisfy $\gcd(a,b)=1$. Residual Artin's conjecture. If the arithmetic progression…

Let $K/\mathbb{Q}$ be a number field with ring of integers $\mathcal O_K$, and let $P$ be a nonzero prime ideal of $\mathcal O_K$. For $g\in K$ with $v_P(g)\geq0$, write…

Let $g\in\mathbb{N}$ be not an exact power, let $\mathcal P$ be the set of primes, and let $\mathcal P_g\subseteq\mathcal P$ consist of the primes $p$ for which…

Let $p$ range over primes, and let $\hat{g}(p)$ denote the least prime primitive root modulo $p$. Let $\gamma$ be the Euler–Mascheroni constant. The least prime primitive root conj…

Let $q$ be a fixed integer with $q\ne 1$ and $q\ne n^2$, and let $p$ be a prime. A number $q$ is a primitive root modulo $p$ when its residue class generates the multiplicative gro…

Let $a>1$ be an integer that is not a perfect square, and let $f_a(p)$ denote the multiplicative order of $a$ modulo a prime $p$. Artin's conjecture. The integer $a$ is a primitive…

A Wieferich prime is a prime $p$ such that $p^2$ divides $2^{p-1}-1$. The Wieferich-prime conjecture. There are infinitely many Wieferich primes, and they are very rare. Wieferich…

Let $p=2r+1$ be a Sophie Germain prime, so that $r$ is also prime, and let $\alpha\in{\mathbb{F}}_{2^r}^*$ be the Gauss period associated with $p$. Gao–Vanstone conjecture. The Gau…

Let $p$ be an odd prime number, and let ${\bf A}(p)$ denote the set of all primitive roots modulo $p$. Golomb's conjecture. There exist $\alpha,\beta\in{\bf A}(p)$ such that … The…

Artin's conjecture. There are infinitely many Artin primes.

Let $p$ be an odd prime, let $g$ be a primitive root modulo $p$, and define $\sigma_g$ on $\mathbb Z/p\mathbb Z=\{0,1,\ldots,p-1\}$ by … for $b\in\{1,\ldots,p-1\}$, with…

Let $q$ be a prime power, let $\mathbb{F}_{q}$ be the finite field of order $q$, and let $a,b,c\in\mathbb{F}_{q}$ satisfy … Let $g_m$ and $g_n$ denote primitive roots of…