Open-problem statements, with their sources attached.

Ecklund, Erdős, and Selfridge [EES74] conjectured $g(k)\leq \exp((1+o(1))k)$.

Erdős, Lacampagne, and Selfridge [ELS93] write 'it is clear to every right-thinking person' that $g(k)\geq\exp(c\frac{k}{\log k})$ for some constant $c>0$.

Sorenson, Sorenson, and Webster [SSWE20] give heuristic evidence that $\log g(k) \asymp \frac{k}{\log k}$.

Is every odd $n > 1$ the sum of a squarefree number and a power of 2?

Erdős often asked this under the weaker assumption that $n > 1$ is not divisible by 4.

Is every odd $n > 1$ the sum of a squarefree number and two powers of 2?

1. There is NO good sequence with polynomial growth.

1. There is a good sequence with sub-exponential growth.

Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of $∏_{i= 1} ^ {n} p(n)$, then $F(n)$ tends to infinity when $n$ tends to infinity.

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

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?

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?

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.

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

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.

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

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).

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."