Open-problem statements, with their sources attached.

Is it true that for every $\epsilon,\eta>0$ there exists a $k$ such that the density of $n$ for which $P(n(n+1)\cdots(n+k))>n^{1-\epsilon}$ is at least $1-\eta$ (where $P(m)$ is the greatest prime divisor of $m$)?

Prove that $F(n)\to \infty$ as $n\to \infty$.

Do infinitely many pairs $(a, a+2)$ occur in Ulam's sequence?

Are there $n$ such that $n+2^{2^k}$ is always squarefree?

Does Ulam's sequence eventually have periodic differences? That is, is $a(n+1) - a(n)$ eventually periodic?

Are there $n$ such that $n+2^{2^k}$ is infinitely often a prime?

Part (iii), is the density of the sequence 0?

Are there $n$ such that $n+2^{2^k}$ is infinitely often squarefree?

When $A =\{a_1 < \cdots\}$ corresponds to the set of primes, it is conjectured that the $\limsup$ of the number of representations $n=\sum_{u\leq i\leq v}a_i$ is infinite.

Let $A\subseteq [1,n)$ be a set of integers such that $(a,b)=1$ for all distinct $a,b\in A$. Is it true that $\sum_{a\in A}\frac{1}{n-a}\leq \sum_{p < n}\frac{1}{p}+O(1)$?

When $A =\{a_1 < \cdots\}$ corresponds to the set of primes, it is conjectured that the set of numbers $n$ that have representations $n=\sum_{u\leq i\leq v}a_i$ has positive upper density.

In [Er80] he claims he "did not state this quite correctly" in [Er77c]. The problem in [Er77c] which Erdős is presumably referring to states that if $n < q_1 < \cdots < q_k\leq m$ is the set of primes in $(n,m]$ then $\sum \frac{1}{q_i-n} < \sum_{p < m-n}\frac{1}{p}+O(1)$.

It is conjectured that if $A =\{a_1 < \cdots\}$ and $g$ counts the number of representations $n=\sum_{u\leq i\leq v}a_i$ such that the sum has at least two terms, then for all $n$ we have $1 \leq g(n)$ for sufficiently large $n$.

Let $G$ be the graph with vertex set those pairs $(x,y)\in \mathbb{N}^2$ with $\mathrm{gcd}(x,y)=1$, in which we join two vertices if the differ in only one coordinate, and there by $\pm 1$.

Is there a path going to infinity on $G$, say $P$, such that for all $(x,y)\in P$ both $\min(x,y)>1$ and at least one of $x$ or $y$ is composite?

The weaker version (only $\min(x,y) > 1$) was solved by C. Stewart via the prime-pair path $(p_k, p_{k+1}) \to (p_{k+1}, p_{k+2})$, as recounted in [Er80]; the compositeness condition forbids those anchors and the question is open.

Find a better lower bound!

In [Er92b] Erdős makes the stronger conjecture (for $a=2$, $b=3$, and $c=5$) that, for any $\epsilon>0$, all large integers $n$ can be written as the sum of distinct integers $b_1<\cdots <b_t$ of the form $2^k3^l5^m$ where $b_t<(1+\epsilon)b_1$.

Find a better upper bound!

Let $k \ne 0$ and $3\leq d_1 < d_2 < \cdots < d_r$ be integers of gcd equal to $1$ such that $\sum_{1 \le i \le r}\frac 1{d_i - 1} \ge 1.$ Can all sufficiently large integers be written as a sum of the shape $\sum_i c_ia_i$ where $c_i \in \{0, 1\}$ and $a_i$ is divisible by $d_i ^ k$ and has only the digits $0, 1$ when written in base $d_i$?

Conjectured by Burr, Erdős, Graham, and Li [BEGL96]

Find the value of the limit of MinOverlapQuotient!

Literature question: Does $A + B$ have positive upper density?