Open-problem statements, with their sources attached.

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

Let $h(n)$ be maximal such that, for any set $A\subseteq \mathbb{N}$ of size $n$, the set$\left\{ \frac{a}{(a,b)}: a,b\in A\right\}$has size at least $h(n)$. Is $h(n) = \Theta(n^{2/3})$?

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.

To prove erdos_539.variants.sq_cube_root it suffices to show $n^{2/3}\ll h(n)$.

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.

From [Er73]: The determination of

will perhaps be not too difficult.

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?

Conjecture 1.* Are there infinitely many practical numbers $m$ such that $h(m) < (\log \log m)^{O(1)}$?

More precisely: does there exist a constant $C > 0$ such that for infinitely many practical numbers $m$, we have $h(m) < (\log \log m)^C$?

Conjecture 2.* Is it true that $h(n!) < n^{o(1)}$? That is, for all $\varepsilon > 0$, is $h(n!) < n^\varepsilon$ for sufficiently large $n$?

Conjecture 3.* Or perhaps even $h(n!) < (\log n)^{O(1)}$?

Erdős offered $250 for a proof or disproof.

Let $s_1 < s_2 < \dots$ be the sequence of squarefree numbers. Is it true that for any $\epsilon > 0$ and large $n$, $s_{n+1} - s_n \ll_\epsilon s_n^\epsilon$?

Let $s_1 < s_2 < \dots$ be the sequence of squarefree numbers. Is it true that $s_{n + 1} - s_n \le (1 + o(1)) \cdot (\pi^2 / 6) \cdot \log (s_n) / \log (\log (s_n))$?

In [Er79] Erdős says perhaps $s_{n+1} - s_n \ll \log s_n$, but he is 'very doubtful'.

[Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.