Unsolved Problems

Let $n\geq 4$. Are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, such that all pairwise distances are integers?...

Let $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\leq d_n$. Furthermore, there are infin...

Let $n_1<n_2<\cdots$ be the sequence of integers which are the sum of two squares. Explore the behaviour of (i.e. find good upper and lower bounds for...

Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Prove that $ \sum_{1\leq n\leq N}d_n^2 \ll N(\log N)^2. $ ...

For every $c\geq 0$ the density $f(c)$ of integers for which $ \frac{p_{n+1}-p_n}{\log n}< c $ exists and is a continuous function of $c$. ", "dif...

Let $f(n)$ count the number of solutions to $n=p+2^k$ for prime $p$ and $k\geq 0$. Is it true that $f(n)=o(\log n)$?...

Let $c_1,c_2>0$. Is it true that, for any sufficiently large $x$, there exist more than $c_1\log x$ many consecutive primes $\leq x$ such that the dif...

Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that $ \lim_{n\to \infty}\frac{a_n}{a_{n-1}^2}=1 $ and $\sum\frac{1}{a_n}\in \mathbb{Q}$. Th...

Let $C>1$. Does the set of integers of the form $p+\lfloor C^k\rfloor$, for some prime $p$ and $k\geq 0$, have density $>0$?...

Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that $ \limsup \frac{a_n}{n}=\infty. $ Is $ \sum_{n=1}^\infty \frac{1}{2^{a_n}} $ transcende...

Is $ \sum_n \frac{\phi(n)}{2^n} $ irrational? Here $\phi$ is the Euler totient function....

Is $ \sum \frac{p_n}{2^n} $ irrational? (Here $p_n$ is the $n$th prime.)...

Let $k\geq 1$ and $\sigma_k(n)=\sum_{d\mid n}d^k$. Is $ \sum \frac{\sigma_k(n)}{n!} $ irrational?...

Let $A\subseteq \mathbb{N}$ be such that $ \lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert \to \infty\textrm{ as }x\to \infty $ and $ \sum_{n\in A...

Let $n\geq 1$ and $f(n)$ be maximal such that for any integers $1\leq a_1\leq \cdots \leq a_n$ we have $ \max_{\lvert z\rvert=1}\left\lvert \prod_{i}(...

Let $A\subseteq \mathbb{N}$ be an infinite set. Is $ \sum_{n\in A}\frac{1}{2^n-1} $ irrational?...

Let $a_1,a_2,\ldots$ be a sequence of positive integers with $a_n\to \infty$. Is $ \sum_{n} \frac{\tau(n)}{a_1\cdots a_n} $ irrational, where $\tau(n)...

Are there infinitely many $n$ such that there exists some $t\geq 2$ and distinct integers $a_1,\ldots,a_t\geq 1$ such that $ \frac{n}{2^n}=\sum_{1\leq...

Let $a_n$ be a sequence of positive integers such that for every sequence of positive integers $b_n$ with $b_n/a_n\to 1$ the sum $ \sum\frac{1}{b_n} $...

Let $a_n$ be a sequence of positive integers such that for every bounded sequence of integers $b_n$ (with $a_n+b_n eq 0$ and $b_n eq 0$ for all $n$) t...

Let $1\leq a_1<a_2<\cdots$ be an increasing sequence of integers. How fast can $a_n\to \infty$ grow if $ \sum\frac{1}{a_n}\quad\textrm{and}\quad\sum\f...

Let $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ be the Fibonacci sequence. Let $n_1<n_2<\cdots $ be an infinite sequence with $n_{k+1}/n_k \geq c>1$. Must $...

Let $P$ be a finite set of primes with $\lvert P\rvert \geq 2$ and let $\{a_1<a_2<\cdots\}=\{ n\in \mathbb{N} : \textrm{if }p\mid n\textrm{ then }p\in...

Let $A(n)=\{a_0<a_1<\cdots\}$ be the sequence defined by $a_0=0$ and $a_1=n$, and for $k\geq 1$ define $a_{k+1}$ as the least positive integer such th...

Let $N\geq 1$. What is the largest $t$ such that there are $A_1,\ldots,A_t\subseteq \{1,\ldots,N\}$ with $A_i\cap A_j$ a non-empty arithmetic progress...

Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p\geq 5$?...

If $G$ is a group then can there exist an exact covering of $G$ by more than one cosets of different sizes? (i.e. each element is contained in exactly...

Is there an infinite Lucas sequence $a_0,a_1,\ldots$ where $a_{n+2}=a_{n+1}+a_n$ for $n\geq 0$ such that all $a_k$ are composite, and yet no integer h...

Let $A=\{n_1<\cdots<n_r\}$ be a finite set of positive integers. What is the maximum density of integers covered by a suitable choice of congruences $...

Let $k\geq 3$. Is there a choice of congruence classes $a_p\pmod{p}$ for every prime $p$ such that all sufficiently large integers can be written as $...

Let $n_1<n_2<\cdots$ be an infinite sequence such that, for any choice of congruence classes $a_i\pmod{n_i}$, the set of integers not satisfying any o...

Let $A\subseteq \mathbb{N}$ be an infinite set and consider the following greedy algorithm for a rational $x\in (0,1)$: choose the minimal $n\in A$ su...

Let $p:\mathbb{Z}\to \mathbb{Z}$ be a polynomial whose leading coefficient is positive and such that there exists no $d\geq 2$ with $d\mid p(n)$ for a...

Let $n\geq 1$ and define $L_n$ to be the least common multiple of $\{1,\ldots,n\}$ and $a_n$ by $ \sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}. $ I...

Let $k\geq 1$ and let $v(k)$ be the minimal integer which does not appear as some $n_i$ in a solution to $ 1=\frac{1}{n_1}+\cdots+\frac{1}{n_k} $ with...

Let $f(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there are no solutions to $ \frac{1}{a}= \frac{1}{b_1}+\cdots+\frac{1}{b_k...

Let $f(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there are no solutions to $ \frac{1}{a}= \frac{1}{b}+\frac{1}{c} $ with di...

For integers $1\leq a<b$ let $N(a,b)$ denote the minimal $k$ such that there exist integers $1<n_1<\cdots<n_k$ with $ \frac{a}{b}=\frac{1}{n_1}+\cdots...

Let $a/b\in \mathbb{Q}_{>0}$ with $b$ squarefree. Are there integers $1<n_1<\cdots<n_k$, each the product of two distinct primes, such that $ \frac{a}...

Does there exist some $c>0$ such that, for any $K>1$, whenever $A$ is a sufficiently large finite multiset of positive integers with $\sum_{n\in A}\fr...

Are there infinitely many solutions to $ \frac{1}{p_1}+\cdots+\frac{1}{p_k}=1-\frac{1}{m}, $ where $m\geq 2$ is an integer and $p_1<\cdots<p_k$ are di...

What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there is a function $\delta:A\to \{-1,1\}$ such that $ \sum_{n\in A}\frac{\delta...

Let $k\geq 3$ and $A\subset \mathbb{N}$ be the set of $k$th powers. What is the order of growth of $1_A^{(k)}(n)$, i.e. the number of representations ...

Let $1\leq m\leq k$ and $f_{k,m}(x)$ denote the number of integers $\leq x$ which are the sum of $m$ many nonnegative $k$th powers. Is it true that $ ...

Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a<b$ nonnegative integers are distinct?...

Let $k\geq 3$ and $f_{k,3}(x)$ denote the number of integers $\leq x$ which are the sum of three nonnegative $k$th powers. Is it true that $ f_{k,3}(x...

Suppose $A\subseteq \mathbb{N}$ is a Sidon set. How large can $ \limsup_{N\to \infty}\frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}} $ be?...

Does there exist a minimal basis with positive density, say $A\subset\mathbb{N}$, such that for any $n\in A$ the (upper) density of integers which can...

Let $A\subseteq \mathbb{N}$ and $D(A)$ be the set of those numbers which occur infinitely often as $a_1-a_2$ with $a_1,a_2\in A$. What conditions on $...

Find the best function $f(n)$ such that every $n$ can be written as $n=a+b$ where both $a,b$ are $f(n)$-smooth (that is, are not divisible by any prim...