13 problems

Let $f\in\mathbb{Z}[x]$ be a quadratic polynomial. Call $f$ stable if every iterate $f^n$ is irreducible over $\mathbb{Q}$, and call $f$ critically infinite if the forward orbit of…

Density conjecture. Through numerical experimentation, the limit $\delta$ exists and is approximately $40\%$. The preceding theorem gives the upper bound $\delta<0.5$, while the as…

Let $\mathcal{P}$ be the set of all primes, and call a prime Wieferich if it satisfies the relevant congruence $N^{p-1}\equiv1\pmod{p^2}$ for the base under consideration. Wieferic…

A prime is regular if it does not divide the class number of the $p$-th cyclotomic field. The question concerns whether there are infinitely many regular primes. Positive-density c…

Let $a>1$ be a fixed integer, and let $(w_n)$ be the sequence defined by … For the generalized Artin problem, take $f(n)=1/n$. Laxton's conjecture. The resulting generalized Artin…

Let $\mathcal{ML}$ be the set of primes $p$ for which the Marques–Lengyel conjecture holds, and let $\mathcal{NMLR}$ be the set of primes for which the rational-center variant fail…

Let $(T,Q)$ be a parameter pair for a recursive sequence, and let the sets of prime divisors of its even-indexed and odd-indexed terms be denoted by $P_{\mathrm{even}}$ and…

For each $d\in\mathcal D$, let $c_d$ be the positive relative density of the set $\mathcal P_d$ in the set of prime numbers, as established by the preceding theorem. Maximal-densit…

Let $K$ be a global field, let $W$ be an infinite finitely generated subgroup of $K^{\times}$, let $F$ be a finite Galois extension of $K$, let $C$ be a union of conjugacy classes…

Let $K=K(n,ll)$ be a totally real number field cyclic over $\mathbb{Q}$ of odd prime degree $n$, with odd class number, with $2$ inert, and with every totally positive unit a squar…

Infinitude and density conjecture. The set $\mathcal{A}$ is infinite. Moreover,

For integers $e,f$ with $1\le e\le f$ and $e+f\ge 3$, let ${\mathcal Q}_w(e,f)$ be the set of primes occurring in a weak Carmichael number of the form $p^eq^f$ or $q^ep^f$ for some…

Let ${\cal P}_2$ be the set of primes $p$ for which $2$ is a primitive root modulo $p$. Consider the primes in ${\cal P}_2$ satisfying … Rodier's conjecture. The density of these p…