Open-problem statements, with their sources attached.

The set of indices $n$ for which a prime gap is followed by a larger or equal prime gap has a natural density of $\frac 1 2$.

The set of indices $n$ for which a prime gap is preceeded by a larger or equal prime gap has a natural density of $\frac 1 2$.

There are infintely many indices $n$ such that the prime gap at $n$ is equal to the prime gap at $n+1$. This is equivalent to the existence of infinitely many arithmetic progressions of length $3$, see erdos_141.variants.infinite_three.

A conjecture by Heath-Brown: The sum of squares of the first $N$ gaps between consecutive primes behaves like $N * (log N)^2$.

Is it true that for all c ≥ 0, the density f c of integers for which (p (n + 1) - p n) / log n < c exists and is a continuous function of c?

Let c₁, c₂ > 0. Is it true that for any sufficiently large x, there exists more than c₁ * log x many consecutive primes ≤ x such that the difference between any two is > c₂?

For every $n>2$ there exist distinct integers $1 ≤ x < y < z$ such that $\frac 4 n = \frac 1 x + \frac 1 y + \frac 1 z$.

Schinzel conjectured (see [Si56]) the generalisation that, for any fixed $a$, if $n$ is sufficiently large in terms of $a$ then there exist distinct integers $1\leq x < y < z$ such that $\frac{a}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$

Let $C > 1$. Does the set of integers of the form $p + \lfloor C^k \rfloor$, for some prime $p$ and $k\geq 0$, have density $>0$?

Let $n_1 < n_2 < \cdots$ be a sequence of integers such that

Is

transcendental?

Is $\sum_{n} \frac{\phi(n)}{2^n}$ irrational? Here $\phi$ is the Euler totient function.

Let $n_1 < n_2 < \dots$ be an arbitrary sequence of integers, each with an associated residue class $a_i \pmod{n_i}$. Let $A$ be the set of integers $n$ such that for every $i$ either $n < n_i$ or $n \not\equiv a_i \pmod{n_i}$. Must the logarithmic density of $A$ exist?

Is $\sum_{n=1}^\infty \frac{p_n}{2^n}$ irrational? Here $p_n$ is the $n$-th prime ($p_1=2, p_2=3, \dots$).

Erdős Problem 252: irrationality of the sum for a given $k$.

For a fixed k ≥ 5, is ∑ σ k n / n! irrational?.

Let $A\subseteq \mathbb{N}$ be such that $\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert \to \infty\textrm{ as }x\to \infty$ and $\sum_{n\in A} \{ \theta n\}=\infty$ for every $\theta\in (0,1)$, where $\{x\}$ is the distance of $x$ from the nearest integer. Then every sufficiently large integer is the sum of distinct elements of $A$.

Let $A\subseteq\mathbb{N}$ be an infinite set. Is

irrational?

Let $a_1 < a_2 < \cdots$ be an increasing sequence such that $\frac{a_n}{n} \to \infty$. Is the sum $\sum_{n}^{\infty} \frac{a_n}{2^{a_n}}$ irrational?

Is $a_n = 2^{2^n}$ an irrationality sequence in the above sense?

Is $n!$ an example of an irrationality sequence?