Open-problem statements, with their sources attached.

Let $F_1=F_2=1$ and $F_{n+1} = F_n + F_{n-1}$ be the Fibonacci sequence. Let $n_1 < n_2 < \dots$ be an infinite sequence with $\frac {n_k}{k} \to \infty$. Must $\sum_k \frac 1 {F_{n_k}}$ be irrational?

Let $P$ be a finite set of primes with $|P| \ge 2$ and let $\{a_1 < a_2 < \dots\}$ be the set of positive integers whose prime factors are all in $P$. Is the sum $\sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]}$ rational?

Let $P$ be a finite set of primes with $|P| \ge 2$ and let $\{a_1 < a_2 < \dots\}$ be the set of positive integers whose prime factors are all in $P$. Is the sum $\sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]}$ irrational?

Is there an infinite Lucas sequence $a_0, a_1, \ldots$ where $a_{n+2} = a_{n+1} + a_n$ for $n \ge 0$ such that all $a_k$ are composite, and yet no integer has a common factor with every term of the sequence?

Let $k\geq 3$. Is there a choice of congruence classes $a_p\pmod{p}$ for every prime $p$ such that all sufficiently large integers can be written as $a_p+tp$ for some prime $p$ and integer $t\geq k$?

If $A ⊆ \mathbb{N}$ is such that $A + A$ contains all but finitely many integers then $\limsup 1_A ∗ 1_A(n) = \infty$.

Let $k\geq2$. Is it true that, for any distinct integers $1 < n_1 < \cdots < n_k$ such that $\sum_{i=1}^k \frac{1}{n_i} = 1$, we must have $\max(n_{i+1} - n_i) \geq 3$?

For all large $N$, there exists a prime $p \in [N, 2N]$ such that $\frac{p+1}{2}$ is also prime.

This is an open conjecture. If true, it would imply erdos_287 for all but at most finitely many exceptions (see erdos_287.variants.prime_conjecture_implies).

Is it true that there are only finitely many pairs of intervals $I_1$, $I_2$ such that

This is still open even if $|I_2| = 1$.

It is perhaps true with two intervals replaced by any $k$ intervals.

Is it true for any $k > 2$ that only finitely many $k$ intervals satisfy this condition?

Is it true that, for all sufficiently large $k$, there exists finite intervals $I_1, \dotsc, I_k \subset \mathbb{N}$ with $|I_i| \geq 2$ for $1 \leq i \leq k$ such that

Let $n\geq 1$ and define $L_n$ to be the least common multiple of $\{1,\ldots,n\}$ and $a_n$ by $\sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}$.

Is it true that $(a_n,L_n)=1$ occurs for infinitely many $n$?

This leads to a heuristic prediction (see for example a preprint of Shiu [Sh16]) of $\asymp\frac{x}{\log x}$ for the number of $n\in [1,x]$ such that $(a_n,L_n)=1$.

In particular, there should be infinitely many $n$, but the set of such $n$ should have density zero. Unfortunately this heuristic is difficult to turn into a proof.

If $A \subset \mathbb{N} has$\sum_{n \in A}\frac 1 n = \infty$, then must$A$ contain arbitrarily long arithmetic progressions?

Is it true that, for every $\varepsilon > 0$, $h(N) = \sqrt N + O_{\varespilon}(N^\varespilon)

Is it true that $N(b) \ll \log \log b$?

Let $\frac a b\in \mathbb{Q}_{>0}$ with $b$ squarefree. Are there integers $1 < n_1 < \dots < n_k$, each the product of two distinct primes, such that $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$?