Open-problem statements, with their sources attached.

What is the size of the largest $A\subseteq\{1, \dots, N\}$ such that there is a function $\delta : A \to \{-1, 1\}$ such that

and

for all non-empty $A'\subsetneq A$.

Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of $∏_{i= 1} ^ {n} p(n)$, $F(n)>n$ for sufficiently large $n$.

Let $c(N)$ be the size of the largest $A\subseteq\{1, \dots, N\}$ such that there is a function $\delta : A \to \{-1, 1\}$ such that

and

for all non-empty $A'\subsetneq A$. What is $\Theta(c(N))$?

Let $r \ge 2$. Is every large integer the sum of at most $r + 1$ many $r$-powerful numbers?

Let $c(N)$ be the size of the largest $A\subseteq\{1, \dots, N\}$ such that there is a function $\delta : A \to \{-1, 1\}$ such that

and

for all non-empty $A'\subsetneq A$. Find the simplest $g(N)$ such that $c(N) = O(g(N)).

For each $k \geq 2$, does the set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right\}$ of all finite sums of distinct factorials contain only finitely many $k$-th powers?

Let $c(N)$ be the size of the largest $A\subseteq\{1, \dots, N\}$ such that there is a function $\delta : A \to \{-1, 1\}$ such that

and

for all non-empty $A'\subsetneq A$. Find the simplest $g(N)$ such that $c(N) = o(g(N)).

Does the set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right\}$ of all finite sums of distinct factorials contain only finitely many powerful numbers?

Let $A \subset \mathbb{N}$ be an additive basis of order 2.

Must there exist $B = \{b_1 < b_2 < \dots\} \subseteq A$ which is also a basis such that $\lim_{k\to\infty} \frac{b_k}{k^2}$ does not exist?

Erdős Problem 1113.* Do there exist Sierpiński numbers that possess no finite covering set of primes?

Erdős and Graham [ErGr80] conjectured that the answer is yes. A negative answer would imply that there are infinitely many Fermat primes.

Erdős Problem 329.* Let A ⊆ ℕ be a Sidon set. How large can lim sup_{N → ∞} |A ∩ {1,…,N}| / N^{1/2} be?

Filaseta–Finch–Kozek conjecture (2008).* Every Sierpiński number is either a perfect power or possesses a finite covering set of primes.

The converse: if the maximum density is 1, then any finite Sidon set can be embedded in a perfect difference set modulo $n > 0$.

Since the consequent is false (due to the counterexamples in [Ha47] and [AlMi25]), this implication is logically equivalent to the statement that the maximum upper density of Sidon sets is NOT 1. Because the maximum upper density problem is still open, the truth value of this implication is also an open research problem.

The Collatz conjecture states that for any positive integer $n$, there exists a natural number $m$ such that the $m$-th term of the sequence is 1.

Does there exist a minimal basis $A \subset \mathbb{N}$ with positive density such that, for any $n \in A$, the (upper) density of integers which cannot be represented without using $n$ is positive?

Let $d_n=p_{n+1}-p_n$, where $p_n$ denotes the $n$th prime. Is it true that $\frac{\max_{n < x}d_{n}d_{n-1}}{(\max_{n < x}d_n)^2}\to 0$ as $x\to \infty$?

Let $A = \{1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \ldots\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to $a + b = c + d$). What is the order of growth of $A$? Is it true that $|A \cap \{1, \ldots, N\}| \gg N^{1/2 - \varepsilon}$ for all $\varepsilon > 0$ and large $N$?

Let $1\leq u_1 < u_2 < \cdots$ be the sequence of integers with at most $2$ prime factors. Is it true that $\limsup_{k \to \infty} \frac{u_{k+1}-u_k}{\log k}=\infty?$

Let $A = \{1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, \ldots\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to $a + b = c + d$). What is the order of growth of $A$? Is it true that $|A \cap \{1, \ldots, N\}| \gg N^{1/2 - \varepsilon}$ for all $\varepsilon > 0$ and large $N$?

Are there infinitely many $n > 2$ such that $n - 2^k$ is prime for all $k \geq 1$ with $2^k < n$?

The only known such $n$ are $4, 7, 15, 21, 45, 75, 105$ (OEIS A039669).