Open-problem statements, with their sources attached.

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.

Erdős Problem 872, part (ii) (strong form): for every $\epsilon > 0$, the game length is at least $(1-\epsilon) \cdot n / 2$ for all sufficiently large $n$.

Status note: the forum thread (April-May 2026) records Shortener strategies giving $L(n) \leq (23/48 + o(1)) \cdot n$ (described in the thread as accepted as correct, with a Lean formalization in progress) and a claimed $L(n) \leq 0.19 \cdot n$, either of which would answer this question negatively under the Prolonger-first convention. Neither is published, so the statement is recorded here as the original Erdős question.

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

Forum-related variant: how small can a maximal primitive subset of $\{2, \dots, n\}$ be? The set of primes in $\{2, \dots, n\}$ is a maximal primitive subset of size $\pi(n)$, and the forum thread asks whether this is the smallest possible for all $n \geq 2$. Equivalently: must every completed play of the saturation game, by both players and regardless of strategy, claim at least $\pi(n)$ elements? (Terminal positions of the game are exactly the maximal primitive subsets.)

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

Let A ⊂ ℕ be an additive basis of order k which is minimal in the sense that if B ⊂ A is any infinite set, then A \ B is not a basis of order k.

Must there exist an infinite B ⊂ A such that A \ B is an additive basis of order k + 1?

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 $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$?

Is the upper density of the set of odd numbers that cannot be expressed as a prime plus two powers of 2 positive?

Are there two finite set of primes $P$ and $Q$ such that

?

Asked by Barbeau [Ba76].

[Ba76] Barbeau, E. J., Computer challenge corner: Problem 477: A brute force program.

Are there infinitely many pairs (m, P) where m ≥ 2 is an integer and P is a set of distinct primes such that the following equation holds: $\sum_{p \in P} \frac{1}{p} = 1 - \frac{1}{m}$?

It is conjectured that the set of primary pseudoperfect numbers is infinite.

Is there some constant $c>0$ such that for every $n\geq 1$ there exists some $\delta_k\in \{-1,0,1\}$ for $1\leq k\leq n$ with $0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < \frac{c}{2^n}?$

Is it true that for sufficiently large $n$, for any $\delta_k\in \{-1,0,1\}$, $\left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert > \frac{1}{[1,\ldots,n]}$ whenever the left-hand side is not zero?

Does there exist a set $A \subseteq \mathbb{N}$ such that $|A \cap \{1, \ldots, N\}| = o((\log N)^2)$ and every sufficiently large integer can be written as $p + a$ for some prime $p$ and $a \in A$?

Can the bound $O(\log N)$ be achieved for an additive complement to the primes? [Guy04] writes that Erdős offered $50 for the solution.

Let $A$ be a set of $n$ positive integers. Does $A$ contain a sum-free set of size at least $\frac n 3 + Ω(n)$, where $Ω(n) → ∞$ as $n → ∞$?

Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$. What is $R(N)$?

Let $G$ be an abelian group of size $N$, and suppose that $A \subset G$ has density $\alpha$. Are there at least $\alpha^{15} N^{10}$ tuples $(x_1, \dots, x_5, y_1, \dots, y_5) \in G^{10}$ such that $x_i + y_j \in A$ whenever $j \in \{i, i+1, i+2\}$?

Note: We interpret indices modulo 5.

Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$. What is $\Theta(R(N))$?