Open-problem statements, with their sources attached.

Estimate $h(n)$ by finding a better upper bound.

Noll and Simmons asked, more generally, whether there are solutions to $q_1! \equiv \dots \equiv q_k! \mod p$ for arbitrarily large $k$ (with $q_1 < \dots < q_k$).

Estimate $h(n)$ by finding a better lower bound.

Is it true that $C(x)=x^{1-o(1)}$?

This is discussed in problem A13 of Guy's collection [Gu04].

Is the limit $F(N)/N$ as $N \to \infty$ irrational?

Pomerance [Po89] gave a heuristic suggesting that this is the true order of growth, and in fact $C(x)= x \exp\left(-(1+o(1))\frac{\log x\log\log\log x}{\log\log x}\right)$.

The problem is to determine the limit of the sequence $\frac{F(N)}{\sqrt{N}}$ as $N \to \infty$.

Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$?

Is it true that in any finite colouring of $\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour?

The conjecture is about the function $f(n)$ which counts the number of solutions to $k\sigma(k)=n$, where $\sigma(k)$ is the sum of divisors of $k$. The first bound is that $f(n)$ grows slower than any power of $n^(\frac{1}{\log\log n})$. The second bound is that $f(n)$ is at most a power of $\log n$.

Any graph on $n$ vertices can be decomposed into $O(n)$ many edge-disjoint cycles and edges.

Part (ii) of Erdős Problem 1060: bound on the number of $k \le n$ with $k \sigma_1(k) = n$.

In [Er71] Erdős suggests that only $n-1$ many cycles and edges are required if we do not require them to be edge-disjoint.

How many (ordered) solutions are there to σ(a) + σ(b) = σ(a + b) with a + b ≤ x? Is it true that this number is asymptotic to c * x for some constant c > 0?

What is the smallest $k$ such that $\mathbb{R}^2$ can be red/blue coloured with no pair of red points unit distance apart, and no $k$-term arithmetic progression of blue points with distance 1?

Erdős asked whether the limiting density f n / n exists and, if so, whether it is irrational.

Seems to be open, as of January 2025.

Estimate $n_k$ by finding a better upper bound.

Let $S \subseteq \mathbb{Z}^3$ be a finite set and let $A = \lbrace a_1, a_2, \ldots \rbrace$ be an infinite $S$-walk, so that $a_{i+1} - a_i \in S$ for all $i$. Must $A$ contain three collinear points?

Are there infinitely many primes $p$ such that $p = 2^k * q + 1$ for some prime $q$ and $k ≥ 0$?

This is mentioned as B46 in Unsolved Problems in Number Theory by Richard K. Guy