6 problems

Sondow–Nicholson–Noe conjecture. For every positive integer $m$ and every integer $n\geq N(m)$,

Let $R(m)$ denote the smallest integer such that, for every $x\geq R(m)$, the interval $\left(\frac{x}{2},x\right]$ contains at least $m$ primes, and let $p_{3m}$ denote the $3m$-t…

Long-run conjecture. Once the numbers get big enough, one will observe considerably longer runs of Ramanujan primes than a fair coin-tossing model would suggest, though never as lo…

Twin Ramanujan-prime asymptotic conjecture. If $\pi_2(x)\to\infty$ as $x\to\infty$, then

Twin Ramanujan-prime ratio conjecture. For all $x\ge10^5$,

Longest-run conjecture.