9 problems

Erdős–Mirsky conjecture. The set $S$ is dense in $\mathbb{R}$.

Hassani's density conjecture. For every $\epsilon>0$ there exists $n_{\epsilon}\in\mathbb N$ such that for all $n>n_{\epsilon}$ we have

Let $F(n)$ be the number of primes in the interval $(n^2,(n+1)^2)$. Hassani's lower-bound conjecture. For every $n\geq 3$, we have … The conjecture was checked computationally for…

Let $n\geq 2$, and arrange the first $n^2$ counting numbers, $1,2,3,\dots,n^2$, in an $n\times n$ square. Sierpiński's Hypothesis H1. Each row contains at least one prime. This 195…

Let $\pi(x)$ denote the number of primes at most $x$, and let $x,y\in\funmathbb{Z}$ satisfy $x,y\geqslant 2$. Hardy–Littlewood's second conjecture. The prime-counting function is s…

Extended tail-bound conjecture. As $x\to\infty$,

Let $\delta_2$ be the constant appearing in lower-bound sieve results for almost primes: if a set has level of distribution $x^{\vartheta-\varepsilon}$ with…

Let $p_n$ denote the $n$th prime and let $d_n=p_{n+1}-p_n$. Erdős's conjecture. … The conjecture predicts arbitrarily large relative decreases and increases between consecutive pri…

Let $p_n$ denote the $n$th prime, let $d_n=p_{n+1}-p_n$, and let $J$ be the set of limit points of the sequence $d_n/\log n$. Erdős's conjecture. … The conjecture predicts that eve…