Open-problem statements, with their sources attached.

Are there infinitely many $q$ for which there is a set $A \subset \mathbb{Z}/q\mathbb{Z}$, $|A| = (\sqrt{2} + o(1))q^{1/2}$, with $A + A = \mathbb{Z}/q\mathbb{Z}$? [Gr24]

Surányi was the first to conjecture that the only non-trivial solution to a!b!=n! is 6!7!=10!.

Do the following exist, for arbitrarily large $n$? An abelian group $H$ with $|H| = n^{2+o(1)}$, together with subsets $A_1, ..., A_n, B_1, ..., B_n$ satisfying $|A_i||B_i| \ge n^{2-o(1)}$ and $|A_i + B_i| = |A_i||B_i|$, such that the sets $A_i + B_i$ are disjoint from the sets $A_j + B_k$ ($j \neq k$)?

NOTE: according to [CKS05, 4.1], the conditions should be $A_i + B_j$ disjoint from $A_j + B_k$ for $i \neq k$. See green_36.variants.cks05.

Is Erdos375Prop true?

Variant using the exact simultaneous double product property from [CKS05, 4.1].

Are there infinitely many $n$ such that ${2n\choose n}$ is coprime to $105$?

Given a natural number N, what is the smallest size of a subset of ℕ that contains, for each d = 1, …, N, an arithmetic progression of length k with common difference d.

Is there some absolute constant $C > 0$ such that

for all $n$?

Asymptotic version: determine the asymptotic behavior of m(N, k) as N grows. The solver should determine what function f : ℕ → ℝ eventually equals (fun N ↦ (m N k : ℝ)).

Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime divisor of

is $p$?

Determine the asymptotic equivalence class (theta) of m(N, k).

Let F(n) := \max\{m + p(m) \mid \textrm{m < n composite}\}\} where $p(m)$ is the least prime divisor of $m$. Is it true that $F(n)>n$ for all sufficiently large $n$?

Determine an upper bound (big O) for m(N, k).

Let F(n) := \max\{m + p(m) \mid \textrm{m < n composite}\}\} where $p(m)$ is the least prime divisor of $m$. Does $F(n) - n \to \infty$ as $n\to\infty$?

Determine a strict upper bound (little o) for m(N, k).

A question of Erdős, Eggleton, and Selfridge, who write that in fact it is possible that this quantity is always at least $n+(1-o(1))\sqrt{n}$

Can we improve the lower bound?

There is a $k$, such that $2 \le k \le n - 2$ and $\binom{n}{k}$ can be the product of consecutive primes infinitely often?

Can we improve the best upper bound?

For all $2 \le k \le n - 2$, can $\binom{n}{k}$ be the product of consecutive primes infinitely often?