Open-problem statements, with their sources attached.

Is it true that for every $n \geq 1$ there is a $k$ such that

Is there an infinite Sidon set $A\subset \mathbb{N}$ such that $\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}$ for all $\varepsilon > 0$?

Does there exists a constant c such that f n - 2 * n ~ c * (n / log n)?

Is it true that $\sum_{n\leq x}t_2(n)\ll \frac{x^2}{(\log x)^c}$ for some $c>0$?

Is it true that, for $k\geq 2$, $\sum_{n\leq x}t_{k+1}(n) =o\left(\sum_{n\leq x}t_k(n)\right)?$

Erdős and Hall conjecture that the sum is $o(x^2/(\log x)^c)$ for any $c<\log 2$.

They ask about the behaviour of $t_{n-3}(n!)$ and also ask whether, for infinitely many $n$, $t_k(n!)< t_{k-1}(n!)-1$ for all $1\leq k < n$.

Is it true that for every $k$ there exists $n$ such that $\prod_{0\leq i\leq k}(n-i) \mid \binom{2n}{n}?$

Let $A \subset \mathbb{F}_2^n$ be a set of density $\alpha > 0$. Does $10A$ contain a coset of some subspace of dimension at least $n - O(\log(1/\alpha))$?

More precisely: does there exist an absolute constant $C > 0$ such that for all $n \geq 1$ and all nonempty $A \subseteq \mathbb{F}_2^n$ with density $\alpha > 0$, the sumset $10A$ contains a coset of some subspace of dimension at least $n - C \log_2(1/\alpha)$?

The sumset $10A$ is defined as $\{a_1 + a_2 + \cdots + a_{10} : a_i \in A\}$, using the pointwise scalar multiplication notation 10 • A where • denotes the iterated addition of a set.

Note: We model $\mathbb{F}_2^n$ as Fin n → ZMod 2, which is an $n$-dimensional vector space over $\mathbb{F}_2$.

Brocard's Problem* Does $n! + 1 = m^2$ have integer solutions other than $n = 4, 5, 7$?

Suppose that $A \subset \mathbb{F}_2^n$ is a set of density $\alpha$. What is the largest size of coset guaranteed to be contained in $2A$?

We phrase this by asking for the exact function $F(\alpha, n)$ giving the maximum dimension of a guaranteed coset.

For what functions $g(N) → \infty$ is it true that $\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}$ implies $\limsup 1_A\ast 1_A(n)=\infty$?

Suppose that $A \subset \mathbb{F}_2^n$ has density $\alpha > 1/2 - C/\sqrt{n}$. Does $A + A$ contain a subspace of co-dimension $O_C(1)$? [Sa11, Question 5.1]

Can one show that $\sum_{n\leq x}g_k(n) \sim c_k x\log x$ for some constant $c_k$?

Suppose that $A \subset \mathbb{F}_2^n$ is a set with an additive complement of size $K$. Does $2A$ contain a coset of codimension $O_K(1)$?

Is it true that there is a constant $c_k$ such that for almost all $n < x$ we have $g_k(n)=c_k\log x+o(\log x)$?

Could $2A$ even contain a coset of codimension $O(\log K)$?

Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$?

Suppose $A, B ⊆ \{1, \dots, N\}$ both have size at least $N^{0.49}$. Must the sumset $A + B$ contain a composite number?

If we only allow the digits $1$ and $2$ then $2^{15}$ seems to be the largest such power of $2$.