Open-problem statements, with their sources attached.

Are there $11$ consecutive primes in arithmetic progression?

Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$?

It is open, even for $k=3$, whether there are infinitely many such progressions.

Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$ for almost all $n$?

Fix a $k \geq 3$. Is it true that there are infinitely many arithmetic prime progressions of length $k$?

Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $\limsup f(n)/n=\infty$?

Let $A$ be a finite Sidon set and $A+A=\{s_1<\cdots<s_t\}$. Is it true that $\frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2 \to \infty$ as $\lvert A\rvert\to \infty$?

A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Are there infinitely many primes in each class?

Is it true that for every $k \geq 1$ we have

for all sufficiently large $N$?

A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave? Erdos conjectured that this tends to infinity.

Does there exist a maximal Sidon set $A\subset \{1,\ldots,N\}$ of size $O(N^{1/3})$?

A question of Erdős, Sárközy, and Sós [ESS94].

A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave? Selfridge conjectured that this is bounded.

Let A be an infinite B₂[2] set. Must liminf |A ∩ {1, ..., N}| * N ^ (- 1 / 2) = 0?

Let $k ≥ 2$. Does there exist a prime $p$ and consecutive intervals $I_0,\dots,I_k$ such that $\prod\limits_{n{\in}I_i}n \equiv 1 \mod n$ for all $1 \le i \le k$?

Estimate $h(n)$ by finding a better upper bound.

Noll and Simmons asked, more generally, whether there are solutions to $q_1! \equiv \dots \equiv q_k! \mod p$ for arbitrarily large $k$ (with $q_1 < \dots < q_k$).

Estimate $h(n)$ by finding a better lower bound.

Is it true that $C(x)=x^{1-o(1)}$?

This is discussed in problem A13 of Guy's collection [Gu04].

Is the limit $F(N)/N$ as $N \to \infty$ irrational?

Pomerance [Po89] gave a heuristic suggesting that this is the true order of growth, and in fact $C(x)= x \exp\left(-(1+o(1))\frac{\log x\log\log\log x}{\log\log x}\right)$.