6 problems

Filz's prime-circle conjecture. For every positive even integer $n$, there exists a permutation $\tau\in S_n$ such that all of these cyclic adjacent sums are prime.

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

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-progression conjecture. There exists $B=B(f_1,\ldots,f_k,A)$ such that

Let $\pi_2(x)$ denote the number of primes $p$ with $p+2$ also prime and $p\leq x$. Twin prime conjecture. The function $\pi_2(x)$ diverges as $x\to\infty$. This is one of the cent…

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…