Open-problem statements, with their sources attached.

Case 2: Does $A + B$ have zero lower density, but positive upper density?

Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $|A| = n$ then $\prod_{a\neq b\in A}(a + b)$ has at least $f(n)$ distinct prime factors. Is it true that $\frac{f(n)}{\log n} \to\infty$?

Erdős says that $f(n) = o(\frac{n}{\log n})$ has never been proved.

Let $k\geq 3$. Can the product of any $k$ consecutive integers $N$ ever be powerful? That is, must there always exist a prime $p\mid N$ such that $p^2\nmid N$?

Erdős [Er82c] conjectures that, if $k$ is fixed, then for all $n$ sufficiently large and all positive integers $m$, there must be at least $k$ distinct primes $p$ such that $p\mid m(m+1)\cdots (m+n)$ and yet $p^2$ does not divide the right hand side.

[Er82c] Erdős, Paul, "Miscellaneous problems in number theory". Congr. Numer. (1982), 25-45.,

In [Er80] Erdős asks whether $\lim_{k \to \infty} (W(k))^{1/k} = \infty$

In [Er81] Erdős asks whether $\frac{W(k+1)}{W(k)} \to \infty$.

In [Er80] Erdős asks whether $W(k)/2^k\to \infty$.

Let $A ⊆ \mathbb{N}$. Let $B ⊆ \mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of two elements from $A$. Is it true that for all $\epsilon > 0$ and large $N$, $|\{1,\ldots,N\} \setminus B| \gg_\epsilon N^{1/2 - \epsilon}$?

Is it possible that $|\{1,\ldots,N\} \setminus B| = o(N^\frac{1}{2})$?

Prove an asymptotic formula for $r_k(N)$, the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression.

Show that $r_k(N) = o_k(N / \log N)$, where $r_k(N)$ the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression.

Find functions $f_k$, such that $r_k(N) = O_k(f_k)$, where $r_k(N)$ the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression.

Prove an asymptotic formula for $r_3(N)$, the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $3$-term arithmetic progression.

Does this imply that

Or

Let $s_1 < s_2 < \cdots$ be the sequence of squarefree numbers. Is it true that, for any $\alpha\geq 0$,

exists?

Is it true that $\sum_{n=1}^\infty(-1)^n\frac{n}{p_n}$ converges, where $p_n$ is the sequence of primes?

Note: In the problem statement, $p_n$ is the $n$-th prime, indexed such that $p_1=2, p_2=3, \ldots$. We 0-index here to reflect how Nat.nth works.

What is the limit $F(N)/N$ as $N \to \infty$?

Erdős Problem 17.* Are there infinitely many cluster primes?