Open-problem statements, with their sources attached.

Open benchmark statement: does an $\mathrm{AME}(11,6)$ state exist?

Is there a sequence $1 \le d_1 < d_2 < \dots$ with density 1 such that all products $\prod_{u \le i \le v} d_i$ are distinct?

Open benchmark statement: does an $\mathrm{AME}(11,10)$ state exist?

Does $f(n)$ miss infinitely many integers?

Open benchmark statement: does an $\mathrm{AME}(12,5)$ state exist?

Is $f$ surjective?

Open benchmark statement: does an $\mathrm{AME}(12,6)$ state exist?

How does $f$ grow?

Open benchmark statement: does an $\mathrm{AME}(12,10)$ state exist?

Does $f$ become stationary at some point?

Open Quantum Problem 35: classify all pairs $(n,d)$ with $n \ge 2$ and $d \ge 2$ for which an $\mathrm{AME}(n,d)$ state exists.

Let $a_1 = 2$ and $a_2 = 3$ and continue the sequence by appending to $a_1, \ldots, a_n$ all possible values of $a_i a_j - 1$ with $i \neq j$. Is it true that the set of integers which eventually appear has positive density?

How can the upper bound be improved?

Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$, all of $n-a$ are prime for all $a\in A$ with $0 < a < n$ and $\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?$

How can the lower bound be improved?

Is it true that, for any $c>1/2$, if $p$ is a sufficiently large prime then, for any $n\geq 0$, there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pmod{p}$?

This is discussed in this MathOverflow question [MathOverflow].

What is the exact value of the constant?

Is it true that limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤?

BMO#1

Let $(a_n)_{n \ge 1}$ and $(b_n)_{n \ge 1}$ be two sequences such that $(a_1, b_1) = (1, 2)$ and

Let q : ℕ → ℕ be a strictly increasing sequence of primes such that q (n + 2) - q (n + 1) ≥ q (n + 1) - q n. Must lim q n / (n ^ 2) = ∞?