Open-problem statements, with their sources attached.

Erdős and Graham [ErGr80] also asked about the difference set $A - A$ and whether this has positive density.

[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

Is $B=\{2^m3^n : m,n\geq 0\}$ an essential component?

In [Ru99] Ruzsa states "The simplest set with a chance to be an essential component is the collection of numbers in the form $2^m3^n$ and Erdős often asked whether it is an essential component or not; I do not even have a plausible guess."

The smallest integer which is unknown to be in $A - A$ is $33$.

Let $A$ be an infinite set such that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$. Is it true that $∑_{n \in A} \frac{1}{n} < \infty$?

It may be true that all or almost all integers are in $A - A$.

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 $m$)?

It may be true that all or almost all integers are in $A - A$.

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.