16 problems

Classification conjecture. A $\Phi_{\kappa}$-sequence $(a_n)_n$ is complete if and only if

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…

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…

Artin's primitive root conjecture. The set of $a$-rooted primes is infinite.

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…

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 $q$ 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 $p$ 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…

Vinogradov's conjecture. For every $\epsilon>0$, for all sufficiently large primes $p$, one has

Let $g^{\square}(p)$ denote the least square-free primitive root modulo a prime $p$. Least square-free primitive root conjecture. For all $p>409$ we have … The preceding explicit b…

Artin's conjecture. There are infinitely many primes $p$ such that $2$ is a generator of the multiplicative group $\mathbb{Z}_p^\times$.

Let $f(x)\in\mathbb{Z}[x]$ produce infinitely many primes, and let $g$ be a square-free integer such that all but finitely many primes produced by $f(x)$ remain inert in…

Let $f(x)\in\mathbb{Z}[x]$ produce infinitely many primes. For an integer $g$, define … Here an Artin prime for $g$ is a prime for which $g$ is a primitive root modulo that prime.…

Let $a,b,r\in\mathbb{Z}$ with $\gcd(a,b)=1$, and suppose that $r\neq -1$ and that $r$ is not a perfect $m$th power for any $m>1$. An integer $r$ is a primitive root modulo a prime…

Matthews' generalized conjecture. The displayed asymptotic holds for the number of primes $\le x$. The source introduces this as a generalized version of Artin's conjecture and use…