Unsolved Problems

Define Ulam's sequence $1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, \ldots$ where $u_1 = 1, u_2 = 2$, and $u_{n+1}$ is the smallest number uniquely ...

Suppose that a large sieve process leaves a set of quadratic size. Is that set quadratic?...

Suppose that a small sieve process leaves a set of maximal size. What is the structure of that set?...

Suppose $A, B \subset \{1, \dots, N\}$ both have size $N^{0.49}$. Does $A + B$ contain a composite number?...

Suppose $A + A$ contains the first $n$ squares. Is $|A| \geq n^{1-o(1)}$?...

Do there exist infinitely many primes $p$ for which $p-2$ has an odd number of prime factors (counting multiplicity)?...

Is there $c > 0$ such that whenever $A \subset [N]$ has size $N^{1-c}$, the difference set $A - A$ contains a nonzero square?...

Let $c > 0$ and let $A$ be a set of $n$ distinct integers. Does there exist $\theta$ such that no interval of length $\frac{1}{n}$ in $\mathbb{R}/\mat...

Let $N$ be large. For each prime $p$ with $N^{0.51} \leq p < 2N^{0.51}$, pick a residue $a(p) \in \mathbb{Z}/p\mathbb{Z}$. Is $\#\{n \in [N] : n \equi...

Sieve $[N]$ by removing half the residue classes mod $p_i$, for primes $2 \leq p_1 < p_2 < \dots < p_{1000} < N^{9/10}$. Does the remaining set have s...

Can we pick residue classes $a_p \pmod p$, one for each prime $p \leq N$, such that every integer $\leq N$ lies in at least 10 of them?...

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

Let $C\geq 0$. Is there an infinite sequence of $n_i$ such that $ \lim_{i\to \infty}\frac{p_{n_i+1}-p_{n_i}}{\log n_i}=C? $ ...

Let $A$ be the set of all odd integers not of the form $p+2^{k}+2^l$ (where $k,l\geq 0$ and $p$ is prime). Is the upper density of $A$ positive?...

Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of 2?...

Let $A$ be an infinite set such that there are no distinct $a,b,c\in A$ such that $a\mid (b+c)$ and $b,c>a$. Is there such an $A$ with $ \liminf \frac...

Let $A\subseteq \mathbb{N}$. Let $B\subseteq \mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of two elements ...

Is it true that $ \sum_{n=1}^\infty(-1)^n\frac{n}{p_n} $ converges, where $p_n$ is the sequence of primes?...

Are there infinitely many primes $p$ such that every even number $n\leq p-3$ can be written as a difference of primes $n=q_1-q_2$ where $q_1,q_2\leq p...

We call $m$ practical if every integer $n<m$ is the sum of distinct divisors of $m$. If $m$ is practical then let $h(m)$ be such that $h(m)$ many divi...

Let $n_1<n_2<\cdots$ be an arbitrary sequence of integers, each with an associated residue class $a_i\pmod{n_i}$. Let $A$ be the set of integers $n$ s...

If $A\subseteq \mathbb{N}$ is such that $A+A$ contains all but finitely many integers then $\limsup 1_A\ast 1_A(n)=\infty$....

Let $h(N)$ be the maximum size of a Sidon set in $\{1,\ldots,N\}$. Is it true that, for every $\epsilon>0$, $ h(N) = N^{1/2}+O_\epsilon(N^\epsilon)? $...

Is there a set $A\subset\mathbb{N}$ such that $ \lvert A\cap\{1,\ldots,N\}\rvert = o((\log N)^2) $ and such that every large integer can be written as...

Let $A\subset\mathbb{N}$ be such that every large integer can be written as $n^2+a$ for some $a\in A$ and $n\geq 0$. What is the smallest possible val...

Find the optimal constant $c>0$ such that the following holds. For all sufficiently large $N$, if $A\sqcup B=\{1,\ldots,2N\}$ is a partition into two ...

Does there exist $B\subset\mathbb{N}$ which is not an additive basis, but is such that for every set $A\subseteq\mathbb{N}$ of Schnirelmann density $\...

Is there an infinite set $A\subset \mathbb{N}$ such that for every $a\in A$ there is an integer $n$ such that $\phi(n)=a$, and yet if $n_a$ is the sma...

Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$ $ \max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\e...

Does every graph on $n$ vertices with $>\mathrm{ex}(n;C_4)$ edges contain $\gg n^{1/2}$ many copies of $C_4$?...

Is $ \sum_{n\geq 2}\frac{1}{n!-1} $ irrational?...

Let $G$ be a chordal graph on $n$ vertices - that is, $G$ has no induced cycles of length greater than $3$. Can the edges of $G$ be partitioned into $...

Does every set of $n$ distinct points in $\mathbb{R}^2$ determine $\gg n/\sqrt{\log n}$ many distinct distances?...

Let $n$ be a sufficently large integer. Suppose $A\subset \mathbb{R}^2$ has $\lvert A\rvert=n$ and minimises the number of distinct distances between ...

Let $k=k(n,m)$ be minimal such that any directed graph on $k$ vertices must contain either an independent set of size $n$ or a transitive tournament o...

For which number theoretic functions $f$ is it true that, for any $F(n)$ such that $f(n)/F(n)\to 0$ for almost all $n$, there are infinitely many $x$ ...

Let $a,b,c\geq 1$ be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form $a^kb^lc^m$ ($k,l,m\ge...

For any $d\geq 1$ and $k\geq 0$ let $P(d,k)$ be the set of integers which are the sum of distinct powers $d^i$ with $i\geq k$. Let $3\leq d_1<d_2<\cdo...

Let $A = \{ \sum\epsilon_k3^k : \epsilon_k\in \{0,1\}\}$ be the set of integers which have only the digits $0,1$ when written base $3$, and $B=\{ \sum...

Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $\lvert A\rvert=n$ then $\prod_{a eq b\in A}(a+b)$ has at least $f(n)$ distinct prime fa...

Let $A\subset\mathbb{R}^2$ be an infinite set which contains no three points on a line and no four points on a circle. Consider the graph with vertice...

We say that $N$ is powerful if whenever $p\mid N$ we also have $p^2\mid N$. Let $k\geq 3$. Can the product of any $k$ consecutive positive integers ev...

Let the van der Waerden number $W(k)$ be such that whenever $N\geq W(k)$ and $\{1,\ldots,N\}$ is $2$-coloured there must exist a monochromatic $k$-ter...

Let $A\subset (1,\infty)$ be a countably infinite set such that for all $x eq y\in A$ and integers $k\geq 1$ we have $ \lvert kx -y\rvert \geq 1. $ D...

If $H$ is bipartite and is $r$-degenerate, that is, every induced subgraph of $H$ has minimum degree $\leq r$, then $ \mathrm{ex}(n;H) \ll n^{2-1/r}. ...

Let $F(k)$ be the number of solutions to $ 1= \frac{1}{n_1}+\cdots+\frac{1}{n_k}, $ where $1\leq n_1<\cdots<n_k$ are distinct integers. Find good est...

Let $k\geq 3$ and $f(k)$ be the supremum of $\sum_{n\in A}\frac{1}{n}$ as $A$ ranges over all sets of positive integers which do not contain a $k$-ter...

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 ...

A finite set $A\subset \mathbb{R}^n$ is called Ramsey if, for any $k\geq 1$, there exists some $d=d(A,k)$ such that in any $k$-colouring of $\mathbb{R...

Find the smallest $h(d)$ such that the following holds. There exists a function $f:\mathbb{N}\to\{-1,1\}$ such that, for every $d\geq 1$, $ \max_{P_d}...