6 problems

Let $k\in\mathbb{N}$ and let $c$ be a positive real number. Write $\mathbb{P}$ for the set of primes. Frantzikinakis's conjecture. For infinitely many pairs…

Fractional-polynomial prime-pair conjecture. There exists $B=B(f,A)$ such that

For $q,m \in \mathbb{N}$, let $p_n$ denote the $n$-th prime and define … for $\mathbf{a} \in \prod_{i=1}^m (\mathbb{Z}/q\mathbb{Z})^\times$. Dickson's conjecture. For any…

Let $q\ge3$ be a positive integer, let $a$ and $b$ be reduced residue classes modulo $q$, and set $h\equiv b-a\pmod q$. Define … Let $\mathfrak{S}_{q,0}$ denote the inclusion–exclu…

Let $p_n$ denote the $n$th prime, and let $\mathcal{P}_{p_n}$ denote the corresponding sieve pattern. Modal multiplicity conjecture. The number $6$ occurs most frequently among the…

First binomial gcd infinitude conjecture. There are infinitely many positive integers $n$ such that